Active control of loudspeakers: An investigation of

Active control of loudspeakers: An investigation of
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Active control of loudspeakers: An investigation of practical applications
Bright, Andrew Paddock; Jacobsen, Finn; Polack, Jean-Dominique; Rasmussen, Karsten Bo
Publication date:
2002
Document Version
Early version, also known as pre-print
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Citation (APA):
Bright, A. P., Jacobsen, F., Polack, J-D., & Rasmussen, K. B. (2002). Active control of loudspeakers: An
investigation of practical applications.
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Active Control of Loudspeakers:
An Investigation of Practical Applications
Andrew Bright
Ørsted·DTU – Acoustic Technology
Technical University of Denmark
Published by: Ørsted·DTU, Acoustic Technology, Technical University of Denmark, Building
352, DK-2800 Kgs. Lyngby, Denmark, 2002
3
This work is dedicated to the memory of
Betty Taliaferro Lawton Paddock Hunt (1916-1999)
and to
Samuel Raymond Bright, Jr. (1936-2001)
4
5
Table of Contents
Preface
9
Summary
11
Resumé
13
Conventions, notation, and abbreviations
15
Conventions....................................................................................................................................... 15
Notation ............................................................................................................................................. 16
Abbreviations .................................................................................................................................... 20
1.
Introduction
1.1.
1.2.
1.3.
2.
23
Active control of loudspeakers .............................................................................................. 28
Organisation of thesis ............................................................................................................ 29
References.............................................................................................................................. 31
Loudspeaker models
33
2.1. Linear models of loudspeakers .............................................................................................. 33
2.1.1.
Electrical dynamics........................................................................................................ 35
2.1.2.
Mechanical dynamics .................................................................................................... 38
2.1.3.
Electro-mechanical transduction.................................................................................... 40
2.1.4.
Acoustical components .................................................................................................. 40
2.1.5.
Acoustic radiation .......................................................................................................... 43
2.1.6.
Linear frequency response ............................................................................................. 44
2.1.7.
Response prediction of loudspeaker mounted in a sealed cabinet ................................. 44
2.1.8.
Response prediction of a loudspeaker mounted in a vented-box enclosure................... 47
2.2. Nonlinear models of loudspeakers......................................................................................... 48
2.2.1.
Parametric nonuniformity and causes of nonlinearity ................................................... 48
2.2.2.
Nonlinear simulation...................................................................................................... 54
2.3. Discrete-time physical modelling .......................................................................................... 56
2.3.1.
FIR filter for electrical admittance................................................................................. 58
2.3.2.
IIR filter for receptance of an SDOF system ................................................................... 60
2.3.3.
IIR filter for mobility of an SDOF system....................................................................... 62
2.3.4.
Nonlinear discrete-time loudspeaker model .................................................................. 64
2.4. Parametric uncertainty ........................................................................................................... 68
2.5. References.............................................................................................................................. 69
3.
Theory of active control of loudspeakers
71
3.1. Feedback control for loudspeakers ........................................................................................ 72
3.1.1.
Constant-current output amplifiers ................................................................................ 72
3.1.2.
Negative amplifier output impedance ............................................................................ 72
3.1.3.
Feedback processing using vibration measurement....................................................... 74
3.2. Feedforward controllers ......................................................................................................... 76
3.2.1.
Linear feedforward processing ...................................................................................... 76
6
3.2.2.
Nonlinear feedforward processing ................................................................................. 76
3.3. Feedback linearisation............................................................................................................ 77
3.3.1.
Feedback linearisation of continuous-time systems....................................................... 77
3.3.2.
Example: simplified closed box loudspeaker in continuous-time.................................. 80
3.3.3.
State observer and partial state measurement ................................................................ 82
3.3.4.
Feedforward formulation using a ‘simulation’ state observer ....................................... 83
3.3.5.
Feedforward formulation assuming ideal alignment..................................................... 84
3.3.6.
Feedback linearisation of discrete-time systems............................................................ 86
3.3.7.
Feedback linearisation with a discrete-time loudspeaker model.................................... 87
3.4. Adaptive feedforward controllers .......................................................................................... 88
3.5. System identification by adaptive filtering ............................................................................ 90
3.5.1.
General adaptive algorithms .......................................................................................... 90
3.5.2.
Adaptive IIR filters ......................................................................................................... 92
3.6. References.............................................................................................................................. 94
4.
Loudspeaker system identification
99
4.1. Overview of approach, implementation, and evaluation ....................................................... 99
4.1.1.
Forms for developing an error equation....................................................................... 100
4.1.2.
Hardware implementation and system-under-test........................................................ 101
4.1.3.
Software implementation ............................................................................................. 104
4.1.4.
Stability triangle........................................................................................................... 106
4.1.5.
Tolerance quadrilateral ................................................................................................ 106
4.1.6.
Frame-based updating .................................................................................................. 110
4.2. Electrical current output-error form ..................................................................................... 112
4.2.1.
Updating Reb ................................................................................................................. 113
4.2.2.
Updating ak................................................................................................................... 114
4.2.3.
Updating of σu as a feedforward coefficient ................................................................ 115
4.2.4.
Updating of σu as a dependent variable, defined by a2 ................................................ 115
4.2.5.
Partial derivative of εoei[n] with respect to φ0............................................................... 116
4.2.6.
Convergence performance ........................................................................................... 117
4.2.7.
Accuracy of the converged parameters for the current output-error form ................... 124
4.2.8.
Updating φk .................................................................................................................. 126
4.3. Voltage output-error form.................................................................................................... 129
4.3.1.
Parameter updating ...................................................................................................... 130
4.3.2.
Convergence performance ........................................................................................... 131
4.4. Displacement equation-error form ....................................................................................... 138
4.4.1.
Displacement from the voltage equation...................................................................... 138
4.4.2.
Displacement from the force equation ......................................................................... 138
4.4.3.
Error definition............................................................................................................. 138
4.4.4.
Parameter updating – linear case ................................................................................. 139
4.5. Conclusions regarding system identification ....................................................................... 143
4.6. References............................................................................................................................ 144
5.
Applications of active control of a loudspeaker
145
5.1. Linear equalisation............................................................................................................... 146
5.1.1.
Acoustic response with equalisation ............................................................................ 146
5.1.2.
Displacement response with equalisation .................................................................... 148
5.2. Compensation of nonlinear distortion.................................................................................. 149
5.2.1.
Simulations of effective sensitivity increase vs. coil height ........................................ 150
5.2.2.
Simulation results......................................................................................................... 155
7
5.2.3.
Distortion compensation on shortened- height voice-coil loudspeakers...................... 159
5.3. References............................................................................................................................ 166
6.
Conclusions
Appendix A. Experimental set-up and tuning
167
169
A.1. Hardware implementation of algorithms ................................................................................. 169
A.2. Electrical impedance measurement ......................................................................................... 169
A.2.1. Differential vs. single-ended considerations................................................................ 169
A.2.2. Extraneous resistances ................................................................................................. 170
A.2.3. Determination of extraneous resistances with single-ended I/O.................................. 172
A.3. Experimental set-up for diaphragm for vibration measurement .............................................. 173
A.4. Experimental set-up for acoustic measurement ....................................................................... 177
Appendix B. Experimental determination loudspeaker parameters
179
B.1. Parameters to be determined.................................................................................................... 179
B.2. Electrical parameter determination (Reb, Leb, and φ0) .............................................................. 179
B.3. Mechanical parameter determination (md, cd, and kd) .............................................................. 182
B.4. References................................................................................................................................ 186
Appendix C. Modal analysis of loudspeaker diaphragms
187
C.1. Introduction.............................................................................................................................. 187
C.2. Modal analysis of structural vibration ..................................................................................... 188
C.3. Experimental modal analysis ................................................................................................... 189
C.4. Interpretation of modal analysis on loudspeakers.................................................................... 190
C.4.1. Measuring structural FRF’s on loudspeakers................................................................ 190
C.4.2. Interpretation of curve-fit parameters .......................................................................... 191
C.4.3. Differential vs. single-ended considerations................................................................ 193
C.5. Summary.................................................................................................................................. 194
C.6. References................................................................................................................................ 194
Appendix D. Rocking modes in single-suspension loudspeakers
197
D.1. Introduction ............................................................................................................................. 197
D.2. Low frequency modes of vibration of a single suspension loudspeaker ................................. 197
D.2.1. Translational vs. rotational modes of vibration ........................................................... 198
D.2.2. ‘Rocking’ modes of diaphragm vibration .................................................................... 198
D.3. Eigenvalue analysis ................................................................................................................. 202
D.4. Problems with rocking modes ................................................................................................. 202
D.4.1. Mechanical tolerances.................................................................................................. 202
D.4.2. Acoustic radiation ........................................................................................................ 202
D.5. References ............................................................................................................................... 203
8
9
Preface
This thesis is the result of a cooperative project between Nokia Research Center of Finland and The
Section of Acoustic Technology at the Ørsted·DTU, Technical University of Denmark, funded by
Nokia Mobile Phones, as a basic study of the electrodynamic loudspeaker. Theoretical work and
academic study was performed at the Section of Acoustic Technology in Lyngby, Denmark.
Experimental work was performed at the Acoustics Laboratories of Nokia Research Center in
Tampere, Finland.
I am grateful to Kaarina Melkas of Nokia Research Center for her support and for her suggestion that I
undertake this project. I am also grateful to Matti Hämäläinen and Leo Kärkkäinen of Nokia Research
Center for their patient acceptance of my sometimes erratic project schedules. I would like to
acknowledge Petri Haavisto and Jukka Saarinen of Nokia Research Center and Nick Courtis and Juha
Backman of Nokia Mobile Phones for supporting this project (or at least not cancelling it – even when
they perhaps should have).
I would like to thank my supervisors at the DTU, Karsten Bo Rasmussen, Jean-Dominique Polack, and
Finn Jacobsen for the guidance, enthusiasm, and direction contributed by them to this project. I am
also grateful to all of the support staff at the Department of Acoustics, for making a friendly
atmosphere that was conducive to research.
I would like to acknowledge the many persons in Nokia Research Center and Nokia Mobile Phones
who together form the ‘Nokia Audio Community.’ I would particularly like to thank John Cozens for
his assistance in preparing experimental hardware, Leo Kärkkäinen again for his explanations of
mathematical concepts, Kaarina Melkas again for providing magnetic field FEM simulation data, and
Matti Kajala for his assistance in C-programming and using sound card drivers. More broadly, this
project has benefited from the wide and varied discussions I have had with a great many persons
within the Nokia Audio Community on aspects of audio, acoustics, and loudspeakers. It is my hope
that this thesis may make some contribution to their work.
The final stages of research and writing of this project were made possible by grants from Nordisk
Forskerutdanningsakademi (NorFA) and from The Nokia Foundation.
I would also like to acknowledge the direct and indirect support from my family, particularly CB and
Mickey, given to me during this project, over distances short and long.
I am grateful to Philips Speaker Solutions for preparing the modified coil-height loudspeaker samples
investigated in this project.
I would like to thank Wolfgang Klippel for technical suggestions to this project, and all those at
Klippel GmbH for their assistance in fine-tuning their measurement system to the small loudspeakers
studied in this project.
Significant indirect support for this project came from Suvi Takkinen and her expanding family.
I am personally indebted to Pontus Veneskoski and Ritva Jääskeläinen for their hospitality during the
final experimental trials of this project.
Finally, love and thanks to Anu, Dori, Katja, Maxi, Paul, Sipe, Sophie & Thierry, and Ulriikka – for
keeping me sane.
10
11
Summary
This thesis investigates applications of active control for loudspeakers. Active control is considered in
this thesis as a tool to simplify the design or increase the performance of an audio reproduction
system. This view has been made possible by recent reductions in the cost of hardware for digital
signal processing. This thesis presents simple processing techniques for active control of
loudspeakers, and shows how they can be used to increase the net sensitivity of electroacoustic
systems.
Authors and inventors have previously promoted active control primarily as a means to improve the
quality of an existing loudspeaker. A somewhat different view is taken in this thesis. Specifically, it
is considered how introducing active control permits an audio reproduction system to more efficiently
or more simply achieve its specified design target.
Practical considerations require active control to be implemented by digital processing. It is assumed
in this thesis that, for active control to reduce cost, its processing algorithms should be no more
complicated than common audio DSP algorithms already in use in a wide array of products. It is
shown in this thesis that suitably simple algorithms for active control are made possible by using
discrete-time models of loudspeaker dynamics. Additionally, the digital filters describing the
loudspeaker dynamics are kept to the same order as common continuous-time models of loudspeaker
dynamics. This achieves simplifications over previously published techniques for digital loudspeaker
processing, which have modelled loudspeaker dynamics by numerical integration of continuous-time
models, or by high-order non-recursive filters.
It is concluded after a review of literature that adaptive feedforward control is the most practical
architecture for active control of loudspeakers. Its key advantages are its ability to tune itself to
changes in a loudspeaker, and to do this without the need for a direct feedback signal, which is
generally expensive and impractical to obtain. A simple feedforward nonlinear controller is developed
by applying the theory of feedback linearisation to the discrete-time loudspeaker model developed.
Simple adaptive algorithms, using the discrete-time loudspeaker model mentioned above, are
presented. These algorithms are used to determine parametric changes in a loudspeaker, known to
occur due to thermal fluctuations, ageing, and other factors. The convergence properties of the
adaptive algorithms are assessed with signals measured on actual loudspeakers.
Two applications of active control are presented. Linear active control, or equalisation as per classical
loudspeaker theory, is discussed for a simple loudspeaker. The benefit of making such equalisation
adaptive, and a simple manner in which this can be done with digital processing, is presented. A
second, nonlinear application of active control is also considered. It is found that nonlinear active
control permits a reduction in the height of a loudspeaker’s voice-coil. A loudspeaker’s sensitivity is
inversely proportional to its moving mass, and as this moving mass is dominated by the voice coil in
some loudspeakers, reduction of its height permits significant increase in sensitivity. Active control
permits compensation of the nonlinear distortion created by the increase in transduction coefficient
nonuniformity caused by this shortening of the voice coil height. Furthermore, it is shown that the
sensitivity increase provided by shortening the voice-coil height is significantly greater than the
additional amplifier output required for compensating the resulting increase in nonlinear distortion.
Measurements of the nonlinear compensation performance with actual loudspeakers are presented.
Conclusions on optimal design of loudspeakers for use with active control are drawn, and suggestions
for further research are given.
12
13
Resumé
Denne afhandling undersøger aktiv kontrol af højttalere. Aktiv kontrol betragtes som et værktøj der
kan forenkle højttalerkonstruktioner eller forbedre højttaleres egenskaber i lydgengivelsessystemer.
Dette er muliggjort af de senere års faldende priser på digitale signalbehandlingssystemer.
Afhandlingen præsenterer forskellige forholdsvis enkle metoder til aktiv kontrol af højttalere, og viser
hvordan de kan bruges til at forøge elektroakustiske systemers følsomhed.
Forskere og opfindere har hidtil primært interesseret sig for aktive metoder som et middel til forbedre
eksisterende højttalertyper egenskaber. Denne afhandling har et lidt andet sigte. Det undersøges
hvordan en højttalerkonstruktion kan modificeres således at der opnås en forbedring af egenskaberne
ved hjælp af et aktivt system.
Aktive kontrolsystemer kan kun realiseres økonomisk ved hjælp af digital signalbehandling. Hvis
aktive metoder skal reducere omkostningerne må det forudsættes at algoritmerne er så enkle som
muligt - og ikke mere komplicerede end de algoritmer til audioformål som allerede anvendes i en lang
række produkter. Det vises at rimeligt enkle signalbehandlingsalgoritmer kan udvikles ved hjælp af
tids-diskrete højttalermodeller. De digitale filtre som beskriver højttalernes dynamiske egenskaber er
begrænset til samme orden som almindelige tids-kontinuerte højttalermodeller. Herved er der opnået
en forenkling sammenlignet med metoder til kontrol af højttalere beskrevet i litteraturen, idet disse
metoder i almindelighed har modelleret højttalernes egenskaber ved hjælp af numerisk integration af
tids-kontinuerte modeller eller med ikke-rekursive filtre af høj orden.
Efter en gennemgang af litteraturen konkluderes det at adaptiv feedforward-kontrol vil være særlig
hensigtsmæssig i forbindelse med aktiv styring af højttalere. Dette hænger sammen med denne
metodes evne til at justere sig ind efter de ændringer der kan forekomme i en højttaler uden brug af et
direkte feedback-signal, som i almindelighed er dyrt og vanskeligt at opnå. Et ret enkelt, ikke-lineært
feedforward-kontrolsystem er udviklet ved at anvende teorien for feedback-linearisering på den tidsdiskrete højttalermodel.
En række adaptive algoritmer, baseret på den tids-diskrete højttalermodel, præsenteres. Disse
algoritmer bruges til at bestemme de parametriske ændringer i en højttaler som vides at forekomme,
bl.a. på grund af termiske fluktuationer og ældningsprocesser. Konvergensegenskaberne er bestemt
med signaler som er målt på forskellige virkelige højttalere.
Afhandlingen undersøger to anvendelser af aktiv kontrol. Lineær aktiv styring af en højttaler
(equalisation) diskuteres. Fordelen ved adaptiv equalisation demonstreres. Ikke-lineær adaptiv styring
bliver også undersøgt, og det konkluderes at en sådan styring gør det muligt at reducere svingspolens
længde. Da højttaleres følsomhed er omvendt proportional med massen af det svingende system, som
domineres af svingspolens masse, følger det at man kan opnå en forøgelse af højttalerens følsomhed.
Dette resulterer imidlertid i øget ikke-lineær forvrængning. Det vises i afhandlingen at denne
forvrængning kan udkompenseres med aktiv kontrol. Endvidere viser det sig at forøgelsen af
følsomheden er større end det ekstra forstærker-output som kræves til signalbehandlingen.
Signalbehandlingssystemets gunstige virkning dokumenteres med målinger på virkelige højttalere.
Afslutningsvis præsenteres konklusioner vedrørende optimal design af højttalere med aktive
kontrolsystemer, ligesom der gives en række forslag til yderligere forskning.
14
15
Conventions, notation, and abbreviations
Conventions
x(t)
x(s)
X(s)
x[n]
x
x ∈ℜN
X
x& (t )
&x&(t )
x ( n ) (t )
xˆ (t )
A real-valued function of continuous-time
The Laplace transform of x(t)
A complex-valued transfer function
A real-valued function of discrete-time
A vector
A real-valued vector with N elements
A matrix
The derivative of x(t) with respect to time.
The second derivative of x(t) with respect to time.
The nth derivative of x(t) with respect to time.
An estimate of x(t)
The Laplace transform of time-dependent quantities is used in this thesis, as given by
∞
x( s ) = ∫ x(t )e − st dt
0
The resulting Laplace transform x(s) is occasionally used in this thesis in the same equation as singlefrequency (harmonic) exponential notation. In this case, we consider the Laplace transform evaluated
at a single frequency. Note that if only a single frequency is considered, the time domain
representation may be recovered using the complex exponential as follows:
{
x(t ) = Re x( s )e −i 2 πft
}
s = − i 2 πf
This is equivalent to stating s = –iω. Readers accustomed to the convention s = jω may consider:
i = – j.
The following sign notation is used for discrete-time difference equations, and their use in discussing
digital filters along with their z-transform:
y[n] = b0 x[n] + b1 x[n − 1] + ... + bM x[n − M ] − a1 y[n − 1] − ... − a N y[n − N ]
M
⇒
H ( z) =
∑b
nz
−n
n =0
N
1+
∑a
nz
−n
n =1
where y[n] is the output, x[n] is the input, bk are the feedforward coefficients, and ak are the feedback
coefficients. This is in contrast to some presentations wherein ak are the feedforward coefficients, and
bk are the feedback coefficients. Also, in some presentations, the ak (feedback) coefficients have the
opposite sign.
16
Notation
In the case of overloaded (multiple use) symbols, definitions of their different uses are separated by a
semicolon (;), and are listed in the order in which they appear in the text.
Some notation used in Appendices C and D is not listed here. All notation appearing in those
appendices and not listed here is defined explicitly in those appendices.
a
ak
Ae(s)
An
b
bk
B, B(x)
c
c0
cd
cn
cp
ct
Cdisp·n
dn
d wk [n]
dˆφ0 [n]
d
e
f
f(s)
fc(t), fc(s), fc[n]
fc·p[n]
f0
fs
f(x(t))
Fma
Fs
g(x(t))
h(t), h[n]
hdt [n]
h Ae (t )
h ∫dt [n]
h(x(t ) )
H(s)
vector of feedback coefficients of a digital filter
kth feedback coefficient of a difference equation, or digital filter
electrical admittance (multiplicative inverse of the electrical impedance)
nth numerator coefficient of a partial-fraction-expansion
vector of feedforward coefficients of a digital filter
kth feedforward coefficient of a difference equation, or digital filter
magnetic flux density (average) in the coil gap created by the magnet in a
loudspeaker, as a function of axial distance along the coil gap
vector of feedback coefficients of a differential equation
speed of sound in air under adiabatic conditions (approx. 343 m/s at 20 °C)
mechanical damping of the loudspeaker diaphragm
nth feedback coefficient of generalised differential equation
effective resistance upon the fluid flow in a port
total mechanical damping of a loudspeaker diaphragm
specified displacement
nth feedforward coefficient of a generalised differential equation
gradient of the error surface ξ(w) along the parameter wk
estimate of the gradient of the error surface along the parameter φ0
vector of feedforward coefficients of a differential equation; vector of gradient
values of the error surface ξ(w) with respect to its weighting parameter vector w
∞
transcendental number = ∑n=01 n!
frequency
force, in frequency (Laplace) domain, generic
mechanical force on the diaphragm, in time (continuous), frequency, and discretetime
force on the voice-coil, predicted from the force equation
the vacuum (acoustically unloaded), open-circuit resonance frequency
free-air resonance frequency of a loudspeaker
feedback (system) vector field of a system represented in state-space form
force due to magnetic attraction (solenoid effect), caused by nonuniform blocked
electrical inductance
sampling rate (frequency) of a discrete-time system ( = 1/ Ts)
input vector field of a system represented in state-space form
impulse response, generic
impulse response of a discrete-time approximation to an differentiator
impulse response of the electrical admittance
impulse response of a discrete-time approximation to an integrator
output vector field of a system expressed in state-space form
transfer function, generic, in continuous-time (Laplace domain)
Conventions, notation, and abbreviations
H X m (z )
i
i(t), i(s)
ic(t), ic(s), ic[n]
ic·m[n]
ic·p[n]
k
k1(x)
ka
kd , kd(x)
kt
l
leff
Leb
L(fn ) h(x(t ) )
ma
md
mt
mp
n
Nφ
Nk
p(t), p(s)
p1m(s)
pb(s)
pc(t)
pf (s)
pr(s)
pσu · n
p
P
Q
Qes
Qms
Qtc
Q
r
rd
Reb
R
17
mechanical mobility, in z-domain (discrete-time)
imaginary number, = − 1
current, generic
electrical current through the voice-coil in time (continuous), frequency, and
discrete-time
measured electrical current
predicted electrical current
integer, generic
deviation from the small-signal stiffness as a function of displacement
stiffness due to acoustic loading on the diaphragm of a loudspeaker
effective stiffness of the diaphragm’s suspension; as a function of displacement
total mechanical stiffness of a loudspeaker diaphragm
effective length of the voice-coil; integer, generic
effective length of the voice-coil
blocked electrical inductance (the effective inductance of the voice-coil when the
voice-coil sees an infinite mechanical impedance)
th
n -order Lie derivative of the vector field h(x(t)) along the vector field f(x(t))
effective mass of acoustic loading on a loudspeaker diaphragm
effective moving mass of a loudspeaker diaphragm
total moving mass of a loudspeaker diaphragm
effective mass of fluid in a port, of a vented (ported) loudspeaker enclosure
integer, generic
order of polynomial fit to the transduction coefficient
order of polynomial fit to the suspension stiffness
acoustic pressure, generic
acoustic pressure at one (1) meter from a compact acoustic source
pressure at back of diaphragm
cavity acoustic pressure
pressure at front of diaphragm
far-field acoustic pressure
nth coefficient of a polynomial approximation of the determination of σu in terms of
a2 (§2.3.3)
new-signal-input-signal cross-correlation vector
matrix for continuous-to-discrete-time feedback coefficient conversion according
to the bilinear transform
‘quality-factor’ of a system’s resonance (= km c )
electrical Q of a loudspeaker
mechanical Q of a loudspeaker
total Q of a loudspeaker mounted in a closed box
matrix for continuous-to-discrete-time feedforward coefficient conversion
according to the bilinear transform
distance from acoustic source to acoustic receiver
effective radius of the loudspeaker diaphragm
blocked electrical resistance (the effective resistance of the voice-coil when the
voice-coil sees an infinite mechanical impedance, or DC electrical resistance)
new-signal autocorrelation matrix
18
s
S0
Sd
Sp
Seff
t
T
Ts
Tc
u(t), u(s)
ud(t), ud(s), ud[n]
up(t), up(s)
ud·f[n]
ud·v[n]
u
v(t), v(s)
vc(t), vc(s), vc[n]
vc·m[n]
vc·p[n]
vu(t)
V0
VAS
w
wopt
xd(t), xd(s)
xd·f[n]
xd·v[n]
Xm(s)
y(t), y[n]
ym(t), ym[n]
yp(t), yp[n]
Ymo(s)
Zeb(s)
Ze(s)
Zmo(s)
Zm(s)
Zrm(s)
Zrmf(s)
Zrmp(s)
Zrmr(s)
Zrad(s)
the ‘Laplace variable’: s = – iω = − 2πf − 1
characteristic sensitivity of a filter
effective area of the loudspeaker diaphragm
effective area of a port
effective sensitivity
time
temperature, in °C
sampling period (interval) of a discrete-time system (=1/Fs)
time constant of a filter = 1/ωc
velocity, generic; control signal output from a controller; input to a system
represented in state-space form
diaphragm velocity
port fluid velocity
diaphragm velocity, simulated from the force equation
diaphragm velocity, simulated from the voltage equation
new-signal vector
voltage, generic; generic input signal to a controller
voltage drop across the terminals of the voice-coil
measured voltage
predicted voltage
voltage generated by velocity of the voice-coil, or ‘back EMF’
volume of a cavity
volume of air with equivalent loading stiffness to suspension stiffness
weight vector of a filter, generic
optimal weight vector (solution to the Wiener filtering problem)
diaphragm displacement as a function of time t, as a function of s in frequency
(Laplace) domain
diaphragm displacement predicted from the force equation
diaphragm displacement predicted from the voltage equation
mechanical mobility
output, generic
output, measured
output, predicted
open-circuit mechanical mobility
blocked electrical impedance
total electrical impedance
open-circuit mechanical impedance, i.e. the mechanical impedance (force/velocity
ratio) of the diaphragm excluding electrical impedance and acoustical loading
general mechanical impedance
mechanical-equivalent acoustic radiation impedance
mechanical-equivalent acoustic radiation impedance on the front of a diaphragm
mechanical-equivalent acoustic radiation impedance of a port radiating into free air
mechanical-equivalent acoustic radiation impedance on the rear of a diaphragm
acoustic radiation impedance for lumped acoustic systems, i.e. the ratio of pressure
to volume velocity of lumped acoustical radiator (note Z rm ( s) = S d2 Z rad ( s) ,
where Sd is the effective area)
Conventions, notation, and abbreviations
α
αk[n]
βk[n]
ε[n]
εoei[n]
ζ
λn
ξ
ξn
ξ(w)
µ
π
πn
ρ0
σu
σx
φ0
φ(x)
φn
χ2
ψ(x)
ψn
ω0
ωz
Ω
∂ a 2 σu (a2 )
19
temperature-resistance coefficient
derivative with respect to coefficient ak
derivative with respect to coefficient bk
error
error of the output-error electrical current algorithm
damping ratio
nth eigenvalue of a system, root of a denominator polynomial, pole in the s-plane
integrand stand-in variable for displacement (x)
nth zero of a discrete-time system, in the z-plane
error-surface function of the weight vector w
convergence parameter, generic
ratio of a circle’s circumference to its diameter
nth pole in the z-plane
density of air under adiabatic conditions (~1.21 kg/m3)
characteristic sensitivity of the discrete-time representation of the mechanical
mobility of an SDOF system
characteristic sensitivity of the discrete-time representation of the mechanical
receptance of an SDOF system
transduction coefficient, under small-signal conditions(=B·l), also referred to as
‘Bl-product,’ or ‘force-factor’
transduction coefficient, displacement-dependent
nth-coefficient of a polynomial approximation to the variation of the transduction
coefficient with respect to diaphragm displacement
chi-squared error function
inverse of the transduction coefficient φ as a function of displacement
nth coefficient of a polynomial approximation to the inverse of the transduction
coefficient
undamped resonance frequency (in radians / sec.)
undamped resonance frequency (in radians / sec.) of a system normalised to the
sampling frequency of a digital system, = 2πf0 / Fs
ohm
∂ φ k u [n]
∂ φ k xd [n]
∂ φ k φ [n]
derivative of σu (characteristic sensitivity of the discrete-time model of the
mechanical mobility) with respect to a2, where σu is defined as a function of a2
derivative of diaphragm velocity ud[n] with respect to φ0 (small-signal transduction
coefficient)
derivative of the diaphragm velocity ud[n] with respect to φk
derivative of φk with respect to xd[n]
derivative of φ(x d [n]) with respect to φk
§
§§
section, e.g. §3.2.1 denotes sub-sub section 1 of sub-section 2 of Chapter 3.
sections, e.g. §§3.1-3.3 denotes to sections 3.1 to 3.3.
∂ φu [n]
20
Abbreviations
BEM
Boundary Element Method
DSP
Digital Signal Processor (Processing)
EMF
Electro-motive force (voltage); specifically as used in discussion of ‘Back-EMF,’ as
created in electrodynamic electro-mechanical transducers.
FEM
Finite Element Method
FIR
Finite impulse response, in reference to digital (discrete-time) filters, also known as a
transversal filter, or tapped delay line filter
FRF
Frequency response function
Hz
Hertz (cycles per second)
IIR
Infinite impulse response, in reference to digital (discrete-time) filters also known as a
recursive digital filter, filter with feedback coefficients, or autoregressive movingaverage (ARMA) model
LHS
Left-hand side
LMS
Least Mean Square (stochastic gradient) adaptive algorithm of Widrow and Hoff
RHS
Right-hand side
SDOF
Single-degree of freedom
Conventions, notation, and abbreviations
21
22
23
1.
Introduction
The loudspeaker is the basic device for generating sound with electronic systems. It is an old, wellknown gadget, dating back to the invention of the telephone in 1875.1 Its construction of magnets and
coiled wire is familiar, sharing much with the rotating electric motor. The increases in inexpensive
computing power in recent decades have spurred interest in the possibility that one might be able to do
with a computer what cannot be done with the dead weight of magnets and wire-wound copper. This
thesis is an investigation into details of several such possibilities.
Considerable research appeared on this subject in the 1990’s. The declared purpose of most such
research was to improve the quality of an existing loudspeaker by digital processing. Specifically, the
most common target has been to reduce nonlinear distortion inherent to a loudspeaker’s mechanical
construction. A slightly different view is taken here. This thesis considers how simple electronic
processing can improve the efficiency, or reduce the size or cost of a given loudspeaker design.
Such a custom designed processor-loudspeaker pair is practical only in the context of an integrated
audio system, and is thus first briefly introduced. Its basic components are shown in Figure 1.1,
comprising a signal processor, power amplifier, and loudspeaker, each of which may be briefly
defined as follows:
Signal Processor: An electronic system for treating or modifying the audio signal in some way.
Power amplifier: A power-electronics system for converting the ‘information’ audio signal from the
processor to a power-drive signal, with sufficient energy to drive the loudspeaker.
Loudspeaker: An electroacoustical transducer generating sound from the electrical power-drive
signal produced by the power amplifier. It should generate sound linearly with respect to the electrical
drive signal, i.e. it should reproduce sound with good ‘fidelity’, so that the information in the audio
signal is reproduced in the sound made by the loudspeaker.
Audio
input signal
Signal
Processor
Power
Amplifier
Loudspeaker
+
-
processed
audio signal
power
audio signal
acoustic
sound field
Figure 1.1: Integrated audio reproduction system.
This ‘integrated audio system’ is used in many different types of commercial products, e.g. telephones,
hearing aids, television sets, integrated ‘hi-fi’ systems and ‘active monitors’ for speech or music
reproduction. As mentioned above, such systems must reproduce the sound field with a certain
fidelity to the audio input signal. Good fidelity reproduction is known to be necessary to maintain
some desired degree of intelligibility of the speech, or quality of the music. Fidelity is not important
1
The first moving-coil (electrodynamic) transducer for audio reproduction was described by Siemens (1874).
However, Siemens’ device did not see widespread use as a loudspeaker as it is known today. It is generally
acknowledged in the loudspeaker engineering community that the first appearance of what is now thought of as
an electrodynamic loudspeaker was first developed by Rice & Kellogg in 1925 (see ref.).
24
Chapter 2
in some types of audio reproductions systems, such as audio alert, warning, or alarm systems. These
are not considered in this thesis.
The cost/benefit analysis of introducing loudspeaker-enhancing signal processing is different for some
types of products using an integrated audio system. Mobile telephones, televisions, and modern
integrated ‘hi-fi’ systems all employ fairly powerful digital computational systems for nonloudspeaker related processing. One example is the mobile telephone, which must employ significant
digital signal processing for speech compression, i.e. coding and de-coding (codec processing). For
such products, the cost of adding loudspeaker-specific processing is much less than other products
without such processors, because the hardware and systems for performing such signal processing are
already present in the product.
Most music signals, such as may be read from an audio compact disc (CD) or received with an FM
radio, will have two or more separate audio signals (channels). Much research has been performed
and continues on signal processing and loudspeaker design for proper reproduction of these ‘multichannel’ audio signals. In this thesis, the focus is on details of the mechanical construction of the
loudspeaker driver1. As these details can be considered in isolation from multi-channel considerations,
only single-channel audio systems (systems for reproducing a single audio input signal) are considered
in this thesis.
Signal Processor
A signal processor may be used in an integrated audio system to treat or modify the audio signal in
some way. As stated above, the purpose of the integrated audio system of Figure 1.1 is to acoustically
reproduce some audio input signal. As the loudspeaker must reproduce this signal with fidelity, the
signal processor may be employed as a servo controller.2
A servo system is considered in two blocks: a controller, and a plant. The controller sends a control
signal u(t) to the plant, based on some processing of the input signal v(t), to ensure that the plant
output y(t) follows the input signal v(t) as closely as possible. For the specific case of the integrated
audio system of Figure 1.1, the servo controller should ensure that the sound field reproduced by the
loudspeaker follows the audio input signal.
An intuitive feature to introduce to such a servo controller is to add closed-loop feedback. Closedloop feedback systems feed the plant output back to the controller, as per the block diagram shown in
Figure 1.2. For the case of the integrated audio system of Figure 1.1, this would mean giving the
controller some feedback about how the loudspeaker is reproducing the audio input signal. Feeding
the output y(t) back to the controller theoretically permits the controller to compensate for problems or
‘errors’ in the plant (the loudspeaker), and thus maintain maximum fidelity between the input v(t) and
the output y(t). If the signal processor can compensate for such problems, it will result in an overall
higher fidelity of acoustic reproduction, improving the quality of the audio reproduction system.
1
The term ‘loudspeaker driver’ is used here to denote a loudspeaker element in isolation from its enclosure.
2
The term servo controller is used here as it is understood within the field of automatic control systems (e.g.
Elgerd, 1967).
Loudspeaker models
Control
signal
Input
v(t)
Controller
u(t)
25
Output
Plant
y(t)
Feedback signal
Figure 1.2: Structure of a feedback, or ‘closed-loop’, control system.
When considering the integrated audio system of Figure 1.1 as a servo system, the signal processor is
part of the controller, and the loudspeaker is part of the plant. The power amplifier, however, may be
either part of the controller, part of the plant, or part of both.1
The advantage of a feedback controller is that it can automatically compensate for various problems in
the loudspeaker. Typical ‘problems’ with loudspeakers are irregular frequency response2 (causing
‘colouration,’ or ‘timbral imbalance’ in the sound), or nonlinear distortion (causing ‘muddiness’ or
‘fuzziness’ and other unpleasant characteristics in the sound). Loudspeaker designers and academics
have studied and experimented with this kind of feedback system for nearly as long as there has been
such a thing that one could call a ‘loudspeaker’ (Adams, 1979).
Feedback controllers for loudspeakers have a significant disadvantage: the output signal y(t) of a
loudspeaker is impractical to obtain. This disadvantage is explained as why no type of closed-loop
controller for a loudspeaker has seen much commercial success, despite some 75 years of research and
The difficulty can be at least
development in academia and industry on such controllers.3
superficially understood by referring back to Figure 1.1. The ‘output’ of this system is the sound field
radiated from the loudspeaker. One could ‘collect’ an output with a microphone at some point in the
sound field, and employ this as a feedback signal. However, due to the nature of sound wave radiation
this feedback signal will be different for different microphone positions. One solution to this problem
could be to place the microphone at the listener’s ear. However, although such a system could
theoretically serve as a good feedback signal, it would be good only for that one user, and only for one
of the user’s ears. The sound field will be different between the user’s right and left ear, and as the
user’s head moves relative to the loudspeaker. Thus correcting the sound for one ear would make it
worse for the other ear, and even worse still for other listeners in the room.
An even more problematic aspect of such a feedback signal is that acoustic propagation from the
loudspeaker to the microphone will introduce a delay, potentially causing the feedback system to fail
basic Nyquist stability criteria, a fundamental measure of the stability of feedback systems. Other
feedback signals have been tried such as measurement of the loudspeaker’s vibration or using a ‘backEMF’ motional-induced signal. None of these systems have met with significant commercial success,
1
Whether the power amplifier is part of the controller, plant, or both depends mainly on the design of the power
amplifier, and whether the signal processor is designed with analogue or digital electronics. The different
possibilities are discussed in detail in Chapter 3, ‘Theory of active control of loudspeakers.’
2
Also referred to as ‘linear distortion.’
3
This problem of motional feedback for loudspeakers is discussed in many papers, e.g. Hanson (1973), Tillett
(1975), Adams (1983), Greiner and Sims (1984), Mills and Hawksford (1989), and Klippel (1992), among
others.
Chapter 2
26
because of either excess cost or poor robustness, or both.1 A more detailed review of these systems is
in Chapter 3 of this thesis.
If the problematic dynamics of the loudspeaker could be known a priori, the controller could simulate
them by an appropriate model, eliminating the need for measurement of an output signal. This type of
processor is referred to as a feedforward or open-loop controller, a block diagram of which is shown in
Figure 1.3. Dispensing the need to measure an output signal from the loudspeaker is a significant
advantage of the feedforward controller. Its disadvantage, however, is that dynamics of the
loudspeaker tend to drift with age, temperature, and other factors. Thus feedforward controllers are
susceptible to parametric misalignment between the controller’s model of the plant and the plantunder-control.
Control
signal
Input
v(t)
Controller
u(t)
Output
y(t)
Plant
Figure 1.3: Structure of a feedforward, or ’open-loop’, control system.
The problem of misalignment between the controller’s a model of the plant and the actual plant can be
solved by system identification. System identification uses the plant model to simulate, in real time,
the plant’s output. Using adaptive filtering, parameters of the model are tuned to minimise the
difference between the output predicted by the model and that measured from the plant. The
‘identified’ parameters of the model can then be used by the feedforward processor, thereby ensuring
the it is properly tuned to the plant. Adding this feature to the feedforward controller results in a
controller known as an adaptive feedforward controller, a block diagram of which is shown in
Figure 1.4. The key advantage of the adaptive feedforward controller over the feedback controller is
that it can use a more indirect output signal from the plant, which is more practical and thus less
expensive to obtain. For this reason, it has been considered the most promising processor
configuration for loudspeakers (Klippel, Nov. 1998), (Klippel, 1999).
Input
v(t)
Controller
Feedforward
Processor
Control
signal
Output
u(t)
y(t)
Plant
ym(t)
yp(t) +
Plant Model
Σ
Plant feedback
System ID by
Adaptive filtering
Figure 1.4: Structure of an adaptive feed-forward control system.
1
This is in contrast to some other applications of motional feedback. One successful applications is for
read/write head arms for disk drives, which use a drive mechanism very similar to that used in a loudspeaker
(some disk drive terminology, such as ‘voice-coil,’ is even borrowed from the loudspeaker industry). The
enabling factor for disk drives seems to come from the fact that position information for the read/write head
arm can be determined directly from formatting information on the disk (Abramovitch and Franklin, 2002).
Loudspeaker models
27
The basic theory of algorithms for updating the parameters of the plant model in the controller, using
recursive adaptive filtering, is presented in Chapter 3.
Power Amplifier
Power amplifiers in modern systems are nearly invariably made from solid-state electronics. These
designs produce amplifiers with low output impedance, otherwise known as constant output-voltage
sources with fixed gain throughout the audio frequency range. Amplifiers employing vacuum-tubes,
or ‘valves’ with considerable output impedance have not completely disappeared, though their
manufacture and use has in the last few decades been relegated to a handful of historical enthusiasts
and musical instrument amplifiers. To date, nearly all power amplifiers have operated on an analogue
input signal. Newer amplifiers, variably referred to as ‘switching’ or ‘Class-D’ amplifiers, have
shown the ability to operate on a digital audio input signal. As these amplifiers also produce a low
impedance output, they may be treated conceptually in the same manner as traditional solid-state
amplifiers.
Other, special types of amplifiers are discussed in the context of motional feedback in Chapter 3.
However, these are included only for the historical background of the application of feedback to
loudspeakers. In all other parts of this thesis, amplifiers are idealised as constant-gain, constant
voltage sources.
Loudspeaker
The idea of a loudspeaker is likely to be familiar to anyone young enough to read this thesis. The
original term comes from loud speaker (two words),1 which was to be contrasted with a ‘regular’
speaker (a telephone receiver). It was ‘loud’ in that sound could be heard without it being held to the
ear. Early loudspeakers (1900-1920) employed small reeds or diaphragms coupled to a horn. The
horn acted as an impedance transformer, matching the high impedance of such mechanical diaphragms
to the low impedance of air. The first widely successful loudspeaker not to use a horn was developed
by Rice and Kellogg (1925). Their invention, known today as a ‘direct-radiator’2 loudspeaker, is
widely recognised as the first appearance of what is understood by a ‘loudspeaker’ today.
Loudspeakers have taken many different forms during their development, a good review of which was
given already half-a-century ago by Beranek (1955).3 It is already clear in Beranek’s mid-century
review that industry at large had settled on the electrodynamic loudspeaker type4. This type of
loudspeaker uses the electrodynamic principle for electro-mechanical transduction, from which it
derives its namesake. Other electro-mechanical transduction principles do exist, and have been used
to construct other types of loudspeakers. A good contemporary review these different transduction
principles is provided by Hixson and Busch-Vishniac (1997). As the electrodynamic transduction
1
Loud Speakers was the title of the first textbook dedicated to loudspeakers. (McLachlan, 1934)
2
Rice and Kellogg’s loudspeaker was a ‘direct’ radiator in the sense that sound was radiated directly from the
diaphragm, without the acoustical aid of a horn. Although horn-loudspeakers remain in use for high-output
loudspeaker systems for public address, nearly all loudspeakers for domestic, portable, and automotive audio
systems use direct-radiator type loudspeakers.
3
It is a sad but generally acknowledged fact that precious little new has been done in loudspeaker design since
the 1950’s. One manifestation of the this appears in monthly patent reviews on loudspeakers by George L.
Auspurger published in the Journal of the Acoustical Society of America, which all too often (accurately) read
‘Every few years someone patents the familiar invention from 195X…’
4
Electrodynamic-type loudspeakers are also referred to as moving coil loudspeakers.
28
Chapter 2
principle remains the only one in widespread use in loudspeakers for audio-frequency reproduction,
the electrodynamic loudspeaker is the only type considered in this thesis. 1
Audio systems which must reproduce the full audio frequency bandwidth typically use two or three
loudspeakers, each for a separate frequency range. This due to the difficulty in reproducing the threedecade2 wide bandwidth of the audio frequency range with a single transducer. This thesis is
concerned with the effects of signal processing on mechanical construction of the loudspeaker, which
can be considered in isolation from multi-transducer loudspeaker systems. Additionally, this thesis
has focused particularly on the ‘microspeaker,’ intended for reproducing speech frequencies, which be
done effectively over a frequency bandwidth of only one decade. For these reasons, only singleloudspeaker systems are considered in this thesis.
1.1. Active control of loudspeakers
The goal of research reported in this thesis has been to find ways, if any, to simplify the mechanical
construction or improve the efficiency of a loudspeaker by electronic signal processing. This is,
broadly speaking, the same goal as other systems employing active control.
The idea of an ‘actively controlled’3 or ‘smart’ loudspeaker has been discussed by several authors.
However, these previous discussions have referred to control of different aspects of the loudspeaker
from those investigated in this thesis. Sophisticated digital systems for controlling high-count
loudspeaker arrays, variably called ‘system controllers’ or ‘loudspeaker processors’ have been in
commercial use for some years (Forsythe et al., 1994). In this thesis, the function of the processor is
considered with regard to how it can simplify the mechanical construction of the loudspeaker.
Active control has shown the ability to produce a more economical solution than passive mechanical
systems in various engineering problems. Two well-known examples are flight control of aerospace
structures and the reduction of acoustic noise and vibration. Active control of aeroplanes, or ‘Fly-bywire’ flight control provides two key advantages over passive-mechanical control systems: the ability
to automatically stabilise an unstable system, and the reduction of weight of a mechanical hydraulicassist system. These two features have seen Fly-by-wire flight control systems serve as a method for
cost-reduction in commercial civilian aircraft (Collinson, 1999). Active control of acoustic noise has
also proven more economical than passive techniques in some applications, such as hearing-protectors
(ear defenders) and propeller-based passenger aircraft (Elliottt, 1999). In both of these successful
fields of active control, electronic processing offers a more economical solution than passive systems,
1
It is generally considered within the Hi-Fi loudspeaker industry that the electrodynamic principle is used by at
least 99% of all loudspeaker elements. One may arrive at a different percentage if one considers that
telephones, personal stereo headphones, automotive audio systems, and portable consumer electronics have all
exclusively used electrodynamic loudspeakers for several decades. As annual production of loudspeakers for
these products runs into the hundreds of millions, the percentage figure is thought to by some to be closer to
99.9999%.
One may wonder whether other known transduction principles have been neglected by force of habit over the
decades. Perhaps the other principles could be used to great effect, but have been forgotten by the mainstream
loudspeaker industry, be it by oversight, ‘technology momentum,’ or vested interest? The truth is that no small
amount of effort has been spent trying to develop loudspeakers based on non-electrodynamic transduction
principles. Any doubter of this fact is referred to Frederick Hunt’s 92-page historical review of
electroacoustics in his 1954 text Electroacoustics (Hunt, 1954).
2
The term ‘decade’ is used here to denote a range of frequencies wherein the highest frequency is 10 times the
lowest frequency. One decade equals approximately 3.322 octaves.
3
The term actively controlled loudspeaker is intended to be different from an active loudspeaker; the latter term
is commonly used to describe loudspeakers with built-in power amplifiers (i.e. in the same cabinet),
Loudspeaker models
29
primarily through weight reduction, achieved by limiting some mechanical bulk needed for the
equivalent passive solution, particularly for aerospace applications. To be sure, these applications of
active control have a significant advantage over the loudspeaker: the cost of a civilian airplane is $40100 million, whereas a loudspeaker averages about $1. This different cost basis for the addition of an
active control system may explain why it has yet to be applied with commercial success to the
loudspeaker.
Of particular interest in battery-powered audio reproductions systems is a loudspeaker’s the pressure
to voltage sensitivity in addition to its efficiency. This is due to the limited voltage output capability
of amplifiers in battery-powered products. Due to the need to lower battery voltages in order to
decrease the power consumption of logic circuits, the voltage available to power amplifiers is limited.
Furthermore, no suitably compact and efficient voltage step-up converters are known to exist which
could solve this problem. Thus in many cases, the acoustic output of audio systems in batterypowered products is limited by the voltage sensitivity of the loudspeaker, and not the power handling
capabilities of the amplifier and loudspeaker.
It was suggested by Klippel (2000), that active control of nonlinear distortion in a loudspeaker be used
to create the same displacement in an electrodynamic loudspeaker, but with a shorter voice-coil height.
Furthermore, it was suggested that the additional output from the amplifier required for the shorter
voice-coil would be modest. This shortening of the voice-coil height can result in a more efficient
loudspeaker in two ways. Firstly, shortening the voice-coil height permits concentration of the coil
wire into the strongest part of the magnetic field, resulting in a higher basic current-to-force
electrodynamic transduction factor1. Secondly, shortening the voice-coil height will reduce the
moving mass of the loudspeaker; as the electrodynamic loudspeaker is a mass-controlled transducer,
this will increase its basic sensitivity. The trade-off between a shorter coil height and the additional
electrical output needed to compensate for the nonlinear distortion created by shortening the coil
height is simulated in detail in Chapter 5.
The general result of these simulations shows that the highest overall-sensitivity is achieved for a coilheight approximately equal to the magnet gap height. This result has been confirmed with a series of
measurements of specially-prepared shortened-coil-height loudspeakers, as reported in Chapter 5.
1.2. Organisation of thesis
Chapter 1. Introduction.
Chapter 2.Loudspeaker models: Classical linear and nonlinear models of the loudspeaker are
reviewed. Application of these models to a miniature electrodynamic loudspeaker, defined as a
microspeaker, is discussed. As active control systems use digital processing techniques that operate
in discrete time, methods for representing the classical linear and nonlinear loudspeaker models in
discrete time are reviewed. A simple discrete-time model of a loudspeaker including its dominant
nonlinearities is developed from established theory. The chapter concludes with a presentation of
causes of parametric variation in these models, leading to parameters which cannot be known a priori.
Chapter 3. Theory of active control of loudspeakers: This chapter presents the theory of signal
processing for active control of loudspeakers. It presents classical and recently published theory in the
context of loudspeakers. The discussion is divided into three categories: (i) feedback processing, (ii)
1
Also known as a transduction coefficient, or B·l-factor; the latter term is most commonly used in the
loudspeaker industry. This term is defined in the context of the general classical models of the loudspeaker in
Chapter 2.
30
Chapter 2
feedforward processing, and (iii) adaptive feedforward processing (as per Figures 1.2 – 1.4 above).
The discussion concludes that the adaptive feedforward processing structure is the most practical for
active control of loudspeakers.
This chapter includes a special section on the theory of feedback linearisation, which is included as an
extended discussion on feedforward processing, i.e. (ii), above. The history of the application of
feedback linearisation to the loudspeaker and how it is used to develop a nonlinear feedforward
controller for active nonlinear control of loudspeakers is presented. In this section a new application
of feedback linearisation to the loudspeaker – a discrete time form – is presented, using the discrete
time loudspeaker model developed in chapter 2. This is used to develop a new, simple algorithm for
nonlinear control of loudspeakers.
Following the conclusion that adaptive feedforward control is the most practical form for active
loudspeaker control, a brief review of adaptive filtering theory is presented, with particular focus on
the LMS algorithm for adaptive recursive (IIR) filters.
Chapter 4. Loudspeaker system identification: This chapter presents implementation of the system
identification part of the adaptive feedforward controller of Figure 1.4. A direct-form LMS IIR output
error algorithm is used to identify those parameters of the discrete-time loudspeaker model which
cannot be known a priori, as discussed in chapter 2. Three different plant model structures are
presented. Equations for iterative parameter identification are derived according to basic adaptive
filtering theory. Convergence performance of the different plant model structures, using data
measured on actual loudspeakers, are presented for different types of signals. Conclusions are drawn
about the most efficient plant model structure for loudspeaker system identification.
Chapter 5. Applications of active control of a loudspeaker: Two applications of active control to a
loudspeaker and their impact on loudspeaker design are considered. Linear control, or equalisation, is
first considered briefly. Nonlinear control, or nonlinear distortion compensation, is considered in
more detail.
The benefit of nonlinear distortion compensation is investigated by simulation. The trade-off between
sensitivity increase from a reduced coil height versus the additional amplifier output required for
compensation of nonlinear distortion caused by the reduced coil height is evaluated. From the
simulation results, the overall sensitivity for different coil heights is considered as a function of coil
displacement, for a range of frequencies.
Based on the simulation results, a special set of modified coil-height loudspeakers were prepared.
Linear and nonlinear measurements on these samples are presented. The ability of the new algorithm
for nonlinear control of loudspeakers, presented in chapter 3, to compensate harmonic distortion
generated in these samples is assessed, and measurements of the reduction in harmonic distortion are
presented. Conclusions are drawn about the optimal coil height of loudspeakers for use with active
control.
Chapter 6. Conclusions: Conclusions are drawn on active control of loudspeakers using adaptive
feedforward processing and how it can benefit loudspeaker design. Potential problems in commercial
implementation are identified. Suggestions for further research are given.
Appendix A. Experimental set-up and tuning: Experimental set-ups used for measurements made in
various parts of the thesis are presented. Diagrams and photographs of the experimental set-ups are
shown. Some techniques used for equipment calibration and parameter tuning are presented.
Loudspeaker models
31
Appendix B. Experimental determination loudspeaker parameters: A method for determining the
linear parameters of a loudspeaker is presented. This provides accurate determination of parameters of
the single-degree-of-freedom loudspeaker model, as well as methods for verification of the model by
reference to measured frequency response functions. Results from this method are used to verify
results from the adaptive algorithms presented in Chapter 4. As this method has not been previously
published, it is included here.
Appendix C. Modal analysis of loudspeaker diaphragms: Techniques for developing multi-degree
of freedom models for loudspeaker diaphragms based on traditional modal analysis for vibrating
systems are presented. The techniques presented here could be used to develop higher-order models
of loudspeaker diaphragm vibration for a feedforward processor, although such techniques are not
presented in this thesis. The technique presented here do present a theoretical framework for
understanding some discrepancies seen between models and measured results.
Appendix D. Rocking modes in single-suspension loudspeakers: Methods for modelling ‘rocking
modes’ in single suspension loudspeakers are presented, based upon techniques for modal analysis of
loudspeaker diaphragms presented in Appendix C. The models of rocking modes describe the
discrepancy between measured and modelled results seen in other parts of the thesis.
1.3. References
Abramovitch, Daniel, and Gene Franklin, “A Brief History of Disk Drive Control,” IEEE Control
Systems Magazine, 22, pp. 28-42. (June, 2002)
Adams, G. J., Optimisation and Motional Feedback Techniques in Loudspeaker System Design, Ph.D.
thesis, The University of Southampton. (Dec. 1979)
Adams, G. J., “Adaptive control of loudspeaker frequency response at low frequencies,” presented at
the 73rd convention of the AES. preprint no. 1983. Journal of the Audio Eng. Soc. (Abstracts) Vol.
31, p 361 (May 1983)
Beranek, Leo L., “Loudspeakers and Microphones”, J. Acoust.. Soc. Amer., 26, pp. 618-629. (Sept.
1955)
Collinson, R. P. G., “Fly-by-wire flight control”, Journal of Computing & Control Engineering, 10,
pp. 141-152. (Aug., 1999)
Elgerd, Olle I., Control Systems Theory, McGraw-Hill Kogakusha, Ltd., Tokyo, Japan. (1967)
Elliott, Stephen J., “Down with Noise”, IEEE Spectrum, pp. 54-61. (June 1999)
Forsythe, Kenton G., Gregory Burlingame, Andrew Rutkin, and Mark Lacas, “New Approaches to
Loudspeaker System Control”, proceedings of the 13th International Conference of the Audio Eng.
Soc., pp. 58-63. (1994)
Greiner, R. A., and T. M. Sims, Jr., “Loudspeaker Distortion reduction,” Journal of the Audio Eng.
Soc., 32, pp. 956-963. (1984)
Hanson, E. R., “A Motional Feedback Loudspeaker System,” Presented at the 46th Convention of the
AES, preprint No. 924. (Sept. 1973)
Hixson, Elmer L., and Ilene J. Busch-Vishniac, “Transducer Principles,” Encyclopaedia of Acoustics,
Chapter 159, John Wiley & Sons, Inc., NY, NY; pp. 1889-1902. (1997)
Hunt, Frederick V., Electroacoustics, Harvard University Press, Cambridge, Mass., USA. (1954)
32
Chapter 2
Klippel, Wolfgang, “The Mirror Filter-A New Basis for Reducing Nonlinear Distortion and
Equalizing Response in Woofer Systems,” Journal of the Audio Eng. Soc., 40, pp. 675-691. (Sept.
1992)
Klippel, Wolfgang J., “Adaptive Nonlinear Control of Loudspeaker Systems,” Journal of the Audio
Eng. Soc. 26, pp. 939-954. (Nov. 1998)
Klippel, Wolfgang J., “Nonlinear Adaptive Controller for Loudspeakers with Current Sensor,”
presented at the 106th Convention of the AES (May 8-11, 1999), preprint no. 4864; Journal of the
Audio Eng. Soc. (Abstracts), 47, p. 512 (Jun. 1999)
Klippel, Wolfgang J., Personal, unwritten correspondence, Paris, France. (February 22, 2000)
McLachlan, N. W., Loud Speakers, Oxford University Press, London, England. (1934)
Mills, P. G. L., and M. O. J. Hawksford “Distortion reduction in moving-coil loudspeaker systems
using current-drive technology.” Journal of the Audio Eng. Soc., 37, pp. 129-148. (Mar. 1989)
Organ, Richard, Avro Arrow, Toronto, Stoddart Publishing Co. Limited, Toronto, Canada. (1992)
Rice, Chester W., and Edward W. Kellogg, “Notes on the Development of a New Type of Hornless
Loud Speaker,” Transactions of the American Institute of Electrical Engineers, 44, pp. 461-475.
(April 1925)
Siemens, Ernst Werner, U. S. Patent No. 149,797 (filed Jan. 20, 1874) issued Apr. 14, 1874.
Tillett, G. W., “Motional Feedback in Loudspeakers,” Audio, 59, pp. 40-43. (Aug. 1975)
Tomayko, James E., “Blind Faith: The United States Air Force and the Development of Fly-By-Wire
Technology,” Technology and the Air Force, U.S. Air Force, Washington, D.C., USA, p. 167.
(1997)
Tomayko, James E., Computers Take Flight: A History of NASA’s Pioneering Digital Fly-by-wire
Project, NASA Dryden Flight Research Center, Edwards, California, USA. (2000)
Loudspeaker models
2.
33
Loudspeaker models
This chapter reviews previously published linear and nonlinear models of loudspeakers. The classical
linear models of the electrodynamic loudspeaker are developed here from first principles. Methods for
introducing nonuniformity of the parameters of the classical linear models, leading to nonlinear
models, are reviewed thereafter.
A study of methods for discrete-time representation of the classical linear models is given. Parametric
variation in these models of the loudspeaker, due to thermal fluctuations and other factors, are
discussed. At the end of the chapter, a nonlinear discrete-time model of the loudspeaker is developed
by using simple continuous-to-discrete-time model conversion methods and the previously published
nonlinear loudspeaker models, with special consideration for inherent discrete-time stability.
2.1. Linear models of loudspeakers
This research has focused specifically on ‘microspeakers;’1 these are small, thin loudspeakers
resembling earpieces or telephone receivers, used in hand-held telephones for generating alert tones
and speech for hands-free telephony. The differences between the ‘microspeaker’ and traditional
direct-radiator electrodynamic loudspeaker are explained in more detail below.
The basic elements of a thin electrodynamic loudspeaker, or ‘microspeaker,’ are shown in Figure 2.1.
A coil of wire, called the ‘voice-coil’ (1), is attached to a diaphragm (2), that is mounted on a fixed
frame (4) via a suspension (3). A magnetic field is generated by a permanent magnet (5) that is
conducted to the region of the coil via a magnetic circuit (6). This generates a concentrated magnetic
field in the region of the coil gap (7). Holes in the rear frame (8) provide ventilation to the rear
enclosure.
According to laws of classical electrodynamics, due to the presence of the magnetic field, electrical
current passing through the voice-coil will generate a force fc in the direction shown in Figure 2.1
(assuming proper orientation of the coil current and magnetic field). This force fc will generate a
displacement xd in the direction shown.
1
The term ‘microspeaker’ is commonly used among producers and telephone manufacturers in North America,
Japan, and the Pacific Rim. The term ‘telecom loudspeaker’ is more common in Europe. Although this thesis
is published by a European institution, the term ‘microspeaker’ is used herein, for brevity, to distinguish this
type of speaker from the traditional direct-radiator electrodynamic loudspeaker.
Chapter 2
34
(1) Voice Coil
(2) Diaphragm
(4) Frame
(7)Magnetic field
in coil gap
Positive
voice-coil
force
+fc
(3) Suspension
(8) Rear-side
ventilation
Positive
coil-diaphragm
displacement
+xd
(5) Magnet
(6) Magnetic
Circuit
10~25mm
Figure 2.1: Cross-section of a thin electrodynamic loudspeaker (‘microspeaker,’) showing its basic components.
The microspeaker is made from the same basic elements as ordinary electrodynamic loudspeakers,
differing primarily in scale and shape. There are some differences worth noting, which are pointed out
here by comparison to a typical low-frequency electrodynamic loudspeaker, an example of which is
shown in Figure 2.2. As shown in Figure 2.2, the voice-coil of a typical loudspeaker is placed on a
former (1b), making a mechanical connection to the diaphragm (2a & 2b). The coil former is typically
a hollow cylinder. This requires that the diaphragm be made in two parts, the outer diaphragm (2a)
and a dust cap (2b), placed over the coil former. The dust cap increases the loudspeaker’s overall
effective radiating area and prevents foreign debris from entering the coil gap. The last difference that
will be noted here is in the suspension; a typical loudspeaker employs two suspensions: an outer
suspension (3a) connecting the diaphragm to the frame (often called the ‘surround’), and an inner
suspension (3b) connecting the coil former to the frame, often called the ‘spider’ due to its spoke
construction in early generations of loudspeakers. The pair of suspensions create a strong stiffness
against ‘rocking’ or ‘wobbling’ of the diaphragm-coil assembly. The absence of this second
suspension in microspeakers makes the microspeaker particularly susceptible to this problem. This
effect can be seen in several measurements presented in this thesis, and is discussed in detail in
Appendix D.
Loudspeaker models
35
(3a) Outer suspension
(surround)
(2a) Diaphragm
(2b) Dust Cap
(3b) Inner suspension
(centering suspension)
(’spider’)
(4) Frame
(1b) Voice Coil
Former
(6) Magnetic
Circuit
(8) Air Vent
(9) ‘Pole piece’
(5) Magnet
(1a) Voice Coil
80~500mm
Figure 2.2: Cross section of a typical low-frequency electrodynamic loudspeaker.
At low displacements of the diaphragm-coil assembly, the electro-mechano-acoustic dynamics are
linear, and thus linear models can be used to describe the relationship between the two.1 At higher
displacement, the relationship is not linear, as parameters of the model vary with displacement.
Distinguishing between ‘high’ and ‘low’ displacement in this context depends on a specific
loudspeaker’s construction. The dominant characteristics are the linearity of the restoring force
provided by the suspension (3) and the uniformity of the force-factor created by the placement of the
voice-coil (1) in the magnetic field (7). Linear models which form part of classical loudspeaker theory
are presented below. The presentation is broken into the electrical (§2.1.1), mechanical (§2.1.2), and
acoustical (§2.1.4) component, with a discussion of the application of these models to the
microspeaker. Models of the parametric nonuniformity that result in nonlinear behaviour are
presented in §2.2, along with measurements of this nonuniformity found in a typical sample of a
microspeaker.
2.1.1. Electrical dynamics
A good, simple model for the electrical behaviour of a loudspeaker is shown in the current loop on the
left-hand side of Figure 2.3.
1
The terms ‘linear’ and ‘nonlinear’ are used here in the algebraic sense, i.e. wherein a linear system obeys the
principles of scalability and superposition, and a nonlinear system does not.
Chapter 2
36
Reb
Leb
ud(t)
ic(t)
vu(t)=φ0ud(t)
vu(t)
vc(t)
fc(t)
Zm(s)
φ0ic(t)=fc(t)
Figure 2.3: Simplified model of electrical elements in an electrodynamic loudspeaker.
The amplifier output is represented by the voltage source vc(t), as nearly all amplifiers used for
loudspeakers are of low output-impedance, constant voltage type. The resistor Reb represents the
electrical resistance of the coil wire. Electrical inductance due to the voice-coil’s shape and
ferromagnetic material in its vicinity is modelled by the inductor Leb. Electro-mechanical transduction
is modelled by a gyrator, shown in the middle of Figure 2.3. This is similar to a transformer, except
that voltage in the primary loop scales with the current in the secondary loop (as opposed to voltage in
the secondary loop for a transformer.) The converse is true for current in the primary loop and voltage
in the secondary loop. The current in the secondary loop, on the right-hand-side of Figure 2.3, is
analogous to the voice-coil velocity ud (t), and therefore the voltage drop induced by the gyrator is the
product φ0ud(t), where φ0 is the gyrator’s constant. All of these effects may be combined into this
single frequency (Laplace) domain equation:
vc ( s ) = (Reb + sLeb )ic ( s ) + φ0ud ( s )
(2.1)
where the terms in (2.1) have the following names:
Voltage drop across the terminals of the voice-coil.
vc(s)
Blocked electrical resistance.
Reb
Blocked electrical inductance.
Leb
φ0
Transduction coefficient (same as ‘B·l-factor,’ or ‘force-factor.’)
Velocity of the diaphragm and coil assembly.
ud(s)
s
The ‘Laplace variable,’ = – iω, where i = − 1 , and ω = 2πf, where f is the frequency in
Hz.
The terms Reb and Leb are the most common description of a loudspeaker’s internal electrical
impedance. Together they describe the electrical impedance when the coil may not move (is
mechanically blocked.) It is thus called the blocked electrical impedance, and represented by the
symbol Z eb (s ) . One may, therefore, generalise (2.1) to
vc ( s ) = Z eb ( s )ic ( s ) + φ0ud ( s )
(2.2)
A phenomenon called eddy currents results in a reactive blocked electrical impedance that differs
significantly from that of a simple inductor. A good model of this phenomenon was developed by
Vanderkooy (1989), wherein the effect was described as an inductance varying with the square-root of
frequency, i.e.:
Im{Z eb } ∝
f
(2.3)
Loudspeaker models
37
Typical examples of the blocked electrical impedance measured from actual loudspeakers are shown
in Figure 2.4 and Figure 2.5. The measurement shown in Figure 2.4 is from a full-range loudspeaker,
and that shown in Figure 2.5 is from a microspeaker. The ‘blocked’ condition (no mechanical
movement) for the full-range loudspeaker has been ensured by placing cyanoacrylate adhesive (‘superglue’) in the coil gap. The measurement for the microspeaker has been synthesised by the difference
between the total electrical impedance and the velocity-induced EMF by a measurement of the velocity,
using a method described in detail in Appendix B.
The reactive (imaginary) part of the impedance of the full-range loudspeaker decreases linearly with
frequency, as it would for a pure inductance, down only to about 200Hz. From 200 to 2000Hz it
varies with f ½, as per the model developed by Vanderkooy, and above 2000Hz it is more or less
invariant with frequency up to the end of the audio frequency range. This strong eddy-current effect
has been intentionally introduced by the designer by attaching a hollow cylinder mounted on the pole
piece (part (9) in Figure 2.2). This has the benefit of reducing the overall blocked electrical
impedance at higher frequencies, thereby increasing the sensitivity at these frequencies (Rausch et al.,
1999)1. By contrast, no such effect is seen in the microspeaker. Its reactive impedance decreases
linearly with frequency, and is thus well-modelled by a simple inductor over the audio frequency
range.
It is also noted that, for the microspeaker, the overall level of the reactive impedance is low relative to
the resistive part over the telephone-band frequency range (up to 4000Hz). This makes it possible, in
some cases, to ignore the effect of the inductance, thereby significantly simplifying some aspects of
modelling the loudspeaker’s dynamics.
Blocked Electrical Impedance − ∅5cm Full range driver
ℜ { Zeb} (Ohms)
20
15
10
5
0
0
5000
10000
15000
10000
15000
ℑ { Zeb} (Ohms)
0
−5
−10
0
5000
Frequency (Hz)
Figure 2.4: Typical blocked electrical impedance of a full-range loudspeaker. Upper: Real part; Lower:
Imaginary part. This loudspeakers .employs a hollow copper cylinder on the pole-piece to reduce the electrical
inductance at high frequency [see Rausch et al. (1999), p. 417, Fig. 8, for a discussion of this effect].
1
see Figures 8 and 10 of Rausch et al. (1999).
Chapter 2
38
Blocked Electrical Impedance − ∅13mm microspeaker
ℜ { Zeb} (Ohms)
10
5
0
0
5000
10000
15000
10000
15000
ℑ { Zeb} (Ohms)
0
−1
−2
−3
−4
−5
0
5000
Frequency (Hz)
Figure 2.5: Typical blocked electrical impedance for a microspeaker. Upper: Real part; Lower: Imaginary part.
The real part is synthesised using the RHS of (B.9), and the imaginary part is synthesised using the RHS of
(B.10), both presented in Appendix B. The aberration appearing at approximately 900Hz is an error caused by
large variations of with respect to frequency in the originally measured frequency response functions from
which these were synthesised.
2.1.2. Mechanical dynamics
A good model of the mechanical dynamics of the electrodynamic loudspeaker is a single-degree-offreedom (SDOF) mechanical oscillator.1 A diagram showing an analogous mechanical system is shown
in Figure 2.6.
to voice
coil
power amplifier
vc
+fc
+xd
φ0
kd
cd
md
Zrm
Figure 2.6: Single-degree-of-freedom (SDOF) mechanical representation of dynamics of an
electrodynamic loudspeaker. The element Zrm is a generalised mechanical impedance, representing
the acoustic loading.
From classical mechanics, the SDOF system is described by this second-order linear inhomogeneous
differential equation:
1
Higher-order models of mechanical dynamics, or multi-degree of freedom (MDOF) models, have also been
studied as part of the research for this thesis. They are not central to the theory of active control as developed
in this thesis, though they are used to explain some differences between modelling and measurement in various
parts of this thesis. For this reason, MDOF models of mechanical dynamics of loudspeakers are discussed in
Appendix C and Appendix D.
Loudspeaker models
md &x&d (t ) + cd x&d (t ) + kd xd (t ) = f c (t )
39
(2.4)
where the terms in (2.4) have the following names:
md
xd(t)
cd
kd
fc(t)
Diaphragm mass
Displacement of the diaphragm.
Damping due to the suspension.
Stiffness (restoring force) due to the suspension.
Force on the voice-coil.
A generalised mechanical impedance Zmo(s) is defined by the Laplace transform of (2.4)
Z mo ( s ) =
fc (s)
= smd + cd + kd s
u d ( s ) ic ( s ) = 0
(2.5)
ps ( s ) =0
where ud(s) is the diaphragm velocity (i.e. the time derivative of xd(s), the diaphragm displacement).
The term Zmo(s) is the in-vacuo open-circuit mechanical impedance. It describes the diaphragm’s force
to velocity ratio under these two conditions:
•
•
The voice-coil is open-circuit ( ic(s)=0 .)
The loudspeaker is in a vacuum, removing acoustical loading ( ps (s) = 0 .)
The term Zmo(s) is also referred to as the ‘internal mechanical impedance.’
The mobility is another important transfer function of an SDOF system, as it more directly characterises
its resonance. It is given by the multiplicative inverse of the impedance as so:
Ymo ( s ) =
1
1
ud ( s )
s
=
=
= 2
f c ( s ) Z mo ( s ) smd + cd + kd s s md + scd + kd
(2.6)
The denominator of this function may be factored as so:
Ymo ( s ) =
1
s
md ( s − λ1 )( s − λ 2 )
(2.7)
where λ1 and λ2 are the roots of the denominator polynomial of (2.6). These define the eigenvalues of
the mechanical system, giving the location of the transfer function’s poles in the s-plane. They are
calculated from the system’s physical parameters according to
λ1 , λ 2 = −ω0ζ ± iω0 1 − ζ 2
(2.8)
where ω0, the undamped natural frequency, is given by
ω0 =
kd
md
(2.9)
and where ζ, the damping ratio, is given by:
ζ=
1 cd
2 kd md
(2.10)
Chapter 2
40
The mobility transfer function of (2.7) may be further separated into first-order terms by partial
fraction expansion, as so:
Ymo ( s ) =
1
1  λ1
λ2 


−
mt λ1 − λ 2  s − λ1 s − λ 2 
(2.11)
This form is particularly convenient for developing an expression for the impulse-response in
continuous-time by inverse Laplace transform, as will be used later in this chapter.
2.1.3. Electro-mechanical transduction
Interaction between the electrical and mechanical components is caused by the classical
electrodynamic interaction of line currents and static magnetic fields. As described in §2.1.1 above,
the forcing term fc(t) on the right-hand-side of (2.4) is produced by a current flowing through a coil in
the magnetic field, which may be modelled by
f c (t ) = B l ic (t )
= φ 0 i c (t )
(2.12)
where ic(t) is the voice-coil current and l is the effective length of the voice-coil wire in a magnetic
field of flux density B. These two quantities are often referred to together as the B·l product. As both
l and B are constants, it is convenient to define their product as the single scalar φ0 called the
transduction coefficient on the urging of Birt (1991). The transduction coefficient φ0 gives a ratio of
force to current, and is therefore also referred to as the force factor.
2.1.4. Acoustical components
The acoustic fluid affects the mechanical behaviour of the loudspeaker. These effects can be modelled
by linear components, assuming the acoustic pressures generated by the diaphragm are accurately
described by the linear acoustic equations. These have the effect of adding terms on the left-hand-side
of the basic diaphragm equation of motions, (2.4). As they are linear operators on xd(t), they may be
separately lumped together, and defined as a single distinct impedance. For this reason, the acoustic
effects on the mechanical behaviour are called the acoustic radiation impedance. Therefore the total
mechanical impedance Zm(s) is defined by
Z m (s) =
f c ( s)
= Z mo ( s ) + Z rm ( s )
ud ( s )
(2.13)
where Zrm(s) is the mechanical-equivalent acoustic radiation impedance. It is related the traditional
lumped-parameter acoustic impedance Zrad(s) by Z rm ( s ) = Sd2 Z rad ( s ) where Sd is the effective radiating
area of the loudspeaker’s diaphragm.
In most applications the loudspeaker is mounted in a cabinet or baffle that prevents acoustic
interaction between the front and rear sides of the diaphragm. This permits the radiation impedance on
the front and rear sides to be treated independently. From the ‘point of view’ of the loudspeaker, the
front and rear loading may be summed together with the internal mechanical impedance as so:
Z m ( s ) = Z mo ( s ) + Z rmf ( s ) + Z rmr ( s ) ,
(2.14)
where Zrmf(s) is the front-side acoustic impedance, and Zrmr(s) is that on the rear-side. Parametric
models of the radiation impedance are developed below.
Closed box enclosure
The simplest rear-acoustic loading commonly used for loudspeakers is a closed-box enclosure.
Mounting a in such an enclosure, as per Figure 2.7, prevents front-to-back sound pressure cancellation,
Loudspeaker models
41
resulting in monopole radiation (instead of dipole radiation, which would be the case without the
enclosure).
Effective diaphragm
radiating area
Sd
Rear volume
Vc
Internal pressure
pc(t)
Figure 2.7: Loudspeaker mounted in a closed-box enclosure
If all dimensions of the cabinet are small compared to the largest wavelength considered, the pressure
is constant throughout the cavity. In this case acoustic pressure p(t) in the cavity is determined by the
changes in the cabinet’s volume caused by movement of the diaphragm. Consider the cavity of
volume V0 shown in Figure 2.8, wherein a pulsating sphere, the volume of which changes according to
V(t).
V(t)
V0
Figure 2.8: Cavity of volume V0 containing a sphere with oscillating volume V(t).
Assuming adiabatic temperature fluctuations, the acoustic pressure in the cavity will be
p (t ) =
V (t )
ρ 0 c 02
V0
(2.15)
The added volume in the cavity V(t) is equal to the product of the effective area Sd and average
displacement –xd(t), where a positive displacement is directed outward from the cavity. Taking the
time derivative of (2.15) and replacing V(t) with –Sdud(t) produces:
dp (t )
S u (t )
= − d d ρ0c02
dt
V0
(2.16)
Rearranging these terms and converting to the Laplace domain gives
pc ( s )
S ρ c2
=− d 0 0
ud ( s )
s Vc
(2.17)
To convert this to a mechanical-equivalent impedance, one should multiply by Sd. Note that a positive
pressure inside the cabinet, on the rear diaphragm, results in a force in the same direction as the
Chapter 2
42
velocity. A positive reactive force resulting from a positive causal velocity denotes a negative
impedance, and thus the effective mechanical impedance produced by the cavity is equal to the
negative of the right-hand-side of (2.17), i.e.
2
Z rmr ( s ) =
S d ρ 0 c02
s Vc
(2.18)
Vented-box enclosure
The addition of a second component to the closed-box enclosure, a lumped mass of air in a small duct
as per Figure 2.9, creates an additional degree of freedom in the system. This system is characterised
by the ratio of port fluid velocity to diaphragm velocity, and the resulting acoustic radiation impedance
presented to the rear side of the driver. This benefits the system response by permitting a reduction in
the cut-off frequency, as well as a reduction in the pressure/displacement ratio. The latter permits
higher acoustic output for the same range of linear displacement.
+ud(t)
Rear volume
Vc
+up(t)
Internal pressure
pc(t)
‘Reflex’ port (vent)
Front area
Spf
Air-slug mass
mp
Rear area
Spr
Figure 2.9: Definition of parameters used to for a single-degree-of-freedom model of ported cavity dynamics
The ratio of port fluid velocity to diaphragm velocity may be determined by first establishing an
equation of motion for the port fluid velocity. From conservation of momentum, or forces, a linear
equation may be derived as so:
(sm
p
)
+ c p + Z rmp ( s ) u p ( s ) = p c ( s ) S p
(2.19)
The terms in (2.19) have the following names:
mp
cp
Zrmp(s)
up(s)
pc(s)
Sp
Mass of the air in the port
Flow resistance of the port
Acoustical impedance seen by the port on the cabinet exterior
Port fluid velocity; positive velocity indicates flow out of the cabinet
acoustic pressure inside the cabinet
Area of the port.
An analytical solution was developed for the radiation impedance of two pistons mounted in an
infinite baffle (Klapman, 1940). As the result does not differ by more than 10% at low k·a from the
single-piston case, and as the impedance contributes less than 10% to the total lumped mass, this effect
is considered negligible in most studies. Therefore the acoustical impedance seen by the port on the
cabinet’s exterior Zrmp(s) may, therefore, be incorporated into proper definition of mp and cp, i.e. by
applying the appropriate ‘end correction.’ It is assumed in the following that this is the case, and thus
the Zrmp(s) term is not used explicitly.
The acoustic pressure inside of the cavity is determined by motion of the diaphragm and fluid flow in
the port according to
Loudspeaker models
pc (s) =
[
]
ρ 0 c 02
− S d u d (s) − S p u p (s) .
sVc
43
(2.20)
The ratio of vent fluid velocity to diaphragm velocity is found by combining Eqs. (2.19) and (2.20), .
which, after some rearrangement, may be expressed as so:
u p ( s)
ud ( s)
=
− S p Sd
ρ 0 c02
sVc
sm p + c p + S p2
(2.21)
ρ 0 c02
sVc
The mechanical equivalent radiation impedance on the rear side of the diaphragm is defined as
Z rmr ( s ) = −
S d pc ( s )
.
ud ( s )
(2.22)
This can be expressed in terms of the physical constants of the system by replacing pc(s) with the RHS
of Eq, (2.20) and then using (2.21) to express up(s) in terms of ud(s). Doing this and rearranging terms
produces:
Z rmr ( s ) =
S d2
ρc 2
sVc
ρ c2

S p2 sV0 c0

1 −

sm p + c p + S p2

ρ 0 c02
sVc





(2.23)
2.1.5. Acoustic radiation
The simplest model for acoustic radiation is
pr ( s ) = sρ0 Sd ud ( s )
eikr
4πr
(2.24)
This model is valid if the shortest wavelength of sound considered is longer than any dimension of the
loudspeaker. The expression in (2.24) is simply the radiation from a point monopole source in free
space, as discussed in elementary acoustics (Pierce, 1994). A feature of monopole radiation important
to loudspeaker design is that the acoustic pressure is directly proportional without frequency
dependence to the volume acceleration of the source. Consequently, in order to produce a frequencyindependent acoustic pressure field, the loudspeaker should produce a constant volume acceleration
with respect to input voltage.
If the loudspeaker is placed in a baffle, in the low k·a region the following will apply:
pr ( s ) = sρ0 Sd ud ( s )
eikr
2πr
(2.25)
This radiation conditions is the most commonly used in the loudspeaker industry. However, for small
products using loudspeakers, e.g. a mobile phone, the 4π (full space) condition is more accurate.
More general models of radiation can be obtained using simulations from boundary element models.
This permits calculation of the radiated pressure from an arbitrarily shaped object, with an arbitrary.
vibration distribution. As the focus of this thesis is on details of the mechanical construction of the
loudspeaker, such methods are not considered in this thesis.
Chapter 2
44
2.1.6. Linear frequency response
It is possible to predict the linear frequency response by combining Laplace domain representations of
the voltage equation, (2.1), and the force equation, (2.4). As most loudspeaker systems are driven by
low-output impedance, constant output voltage amplifiers, the loudspeaker’s response is defined by
the frequency response function referenced to the voice-coil voltage, vc(s). The first step in
developing this transfer function is to combine the transduction equation, (2.12), with the general force
equation, (2.13). This provides this expression for the current in terms of the velocity:
ic ( s ) =
1
ud ( s ) Z m ( s )
φ0
(2.26)
Substituting this into the voltage equation (2.1) and solving for the ratio u d ( s) v c ( s) provides:
ud ( s )
φ0
=
vc ( s ) Z eb ( s ) Z m ( s ) + φ02
(2.27)
Recall that Zm(s) is the sum of the internal mechanical and acoustical load impedances. Thus (2.27)
describes the velocity to voltage ratio including the effects of acoustical loading.
From (2.24), the monopole free-field acoustic radiation depends directly on the diaphragm velocity.
Therefore for the case of a loudspeaker mounted in a closed box, at low k·a numbers, the acoustic
pressure may be determined by combining Eqs. (2.27) and (2.24) to give the ratio of pressure to input
voltage
prp ( s )
ikr
φ0
e p
= sρ0 Sd
vc ( s )
Z eb ( s ) Z m ( s ) + φ02 4πrp
(2.28)
Further characterisation of this formula is given in §2.1.7 below.
The total electrical impedance is a function that will be referred to many times in this thesis. Its
Laplace-domain transfer function may be determined in a similar manner to the pressure response,
above. By first combining the basic definition of mechanical impedance and the relationship between
current and force given in (2.12), the velocity may be written in terms of the current according to
ud ( s ) =
φ0ic ( s )
Z m ( s)
(2.29)
Substituting this into the voltage equation and solving for the ratio v c ( s) i c ( s) provides
v c ( s)
φ 02
Z e ( s) =
= Z eb ( s ) +
i c ( s)
Z m (s)
(2.30)
2.1.7. Response prediction of loudspeaker mounted in a sealed cabinet
Equation (2.28) describes the frequency response function of pressure / voltage in terms of the general
electrical, mechanical, and acoustic radiation impedances of a given loudspeaker and the enclosure in
which it is mounted. It does not, however, directly answer the question of how a loudspeaker and its
cabinet should be designed in order to provide a certain desired response. General answers to the
question of ‘how to design a loudspeaker to achieve some desired response’ were the subject a paper
Loudspeaker models
45
by Thiele (1961) for vented-box loudspeakers,1 along with a different presentation and extension of
Thiele’s work in a series of papers by Small (1971, 1972, Jan. 1973, Jun. 1973, Jul. 1973, Sep. 1973,
and Oct. 1973).
It may be shown that if the loudspeaker is mounted in a sealed enclosure as per Figure 2.7, given
certain assumptions, the transfer function has the form of a second-order high-pass filter. There are
many theorems and laws for filter design that can be used to analyse the loudspeaker’s response. First,
however, it is necessary to re-write the expression of (2.28) into the same form as general high-pass
filters are expressed. The papers by Thiele and Small mentioned above give connections between
physical parameters of a loudspeaker (with its enclosure) and the parameters of high-pass filters. With
this connection, the parameters of the high pass filter (defining the shape of the frequency response)
can be directly known from the physical dimensions and manufacturing specifications of the
loudspeaker.2
Following a method used by Thiele (1961) for vented-box loudspeakers, it was shown by Small (1971)
that the response of a loudspeaker mounted in a closed-box, can be represented by this second-order
high-pass filter:
H hp ( s ) = S0
s 2Tc2
1 + sTc Q + s 2Tc2
(2.31)
where
s = Laplace variable
S0 = system gain
Q = Q-factor
Tc = System time constant = 1 ω c = 1 2πf c , where fc is the system cut-off frequency.
The first step to write the voltage to pressure transfer function of (2.28) into in this form is to
parameterise the general electrical, mechanical, and acoustic radiation impedances (Zeb, Zmo, and Zrm
respectively), as so:
Z eb = Reb + sLeb
Z mo = smd + cd + kd s
2
(2.32)
2
Z rm = sma + ρc S d Vc s
It is helpful to define a single expression for the total mechanical impedance, equal to the sum of the
diaphragm and mechanical-equivalent acoustical impedance:
Z m = Z mo + Z rm = smt + ct + kt s
(2.33)
Here, a total mass and stiffness have been defined as
k t = k d + S d2 ρc 02 Vc
mt = m d + m a
(2.34)
1
The description of the acoustic response of a vented-box loudspeaker as a fourth-order high-pass filter was first
given by Novak (1959). However, Thiele’s 1961 paper was the first to fully apply existing tools for active
analogue filter design to vented box design.
2
Although such a connection may not seem novel from a scientific point of view, Thiele and Small’s work
proved significantly useful to the loudspeaker engineering community. One indication of this is that the basic
set of performance parameters for an electrodynamic loudspeaker, in isolation from its enclosure, are
commonly referred to as the ‘Thiele-Small parameters.’
Chapter 2
46
In addition to the parameterisations of (2.32), it is necessary to assume that the effect of internal
electrical inductance is insignificant, i.e. Leb = 0, so that Zeb ≅ Reb. Notice that, as per the measurement
of the microspeaker shown in Figure 2.5, the net contribution of the electrical inductance to the total
blocked electrical impedance is less than 10% below 5000Hz, and thus this assumption is reasonably
justified, particularly at frequencies below 5000Hz.
If the acoustic pressure is considered at one (1) metre from the loudspeaker in a fully free (4π
steradian) acoustic field, ignoring the phase lag caused be the wave propagation over the one meter
distance, the acoustic propagation term can be simplified from eikr 4πr to 1/4π.
Acoustic response
With the above simplifications and by substituting Eqs. (2.32) and (2.33) into (2.28), it is possible to
express the voltage to pressure transfer function in the second-order low-pass filter form of (2.31) as
so:
p1m ( s ) ρ 0 φ 0 S d
s2
=
v c ( s)
4π Reb m t s 2 + s c t + φ 02 R eb + k t
(
)
(2.35)
This may be described by a gain factor and a generic second-order high-pass filter as so:
p1m ( s )
v( s )
= S 0 H cb ( s )
H pv ( s ) =
= S0
(2.36)
s
s
2
ω 02
2
ω 02
+ s ω 0 Qtc + 1
where
ω0 =
Qtc =
S0 =
kt
mt
(undamped resonance frequency)
kt mt
ct + φ02 Reb
ρ0 φ0 S d
4π m t R eb
(total Q value)
(characteristic sensitivity)
Electrical impedance
The electrical impedance can be represented in a similarly compact form. For this purpose, it is
necessary to define two different Q values. Using notation from Small (1972), these are:
Qes: Electrical Q-factor; this is the Q-factor due to the effective damping provided by a constant
output-voltage amplifier.
Qes =
k t mt
φ 02
Reb
(2.37)
Qms: The mechanical Q-factor; this is the Q-factor due to the damping provided by mechanical
resistance and resistive acoustic radiation impedance.
Qms =
kt mt
ct
(2.38)
Loudspeaker models
47
where ct is the total damping as defined in (2.33).
With these definitions, the general electrical impedance of (2.30) may be expressed as this secondorder narrow-bandpass filter:
s2
s
+
+1
2
ω
ω0
0Qtc
Z e ( s ) = Reb 2
s
s
+
+1
2
ω0 ω0Qms
(2.39)
The magnitude of the frequency response of the total electrical impedance as represented by the
transfer function of (2.39) is plotted in Figure 2.10 for a typical set of values of ω0, Reb, Qtc, and Qms.
30
Impedance (Ω, Mag.)
25
20
15
10
5
0
2
3
10
10
Frequency (Hz)
4
10
Figure 2.10: Plot of parameterised total electrical impedance, for a typical set of values of a microspeaker.
2.1.8. Response prediction of a loudspeaker mounted in a vented-box enclosure
The far-field acoustic pressure generated by a loudspeaker in a vented-box enclosure is caused by the
sum of the volume velocity from the diaphragm and the port. If monopole radiation in a fully free
field is considered, the acoustic pressure at one (1) metre, ignoring propagation delay, will be given by
the diaphragm and port velocities according to
p1m ( s ) =
sρ 0
S d u d (s) + S p u p (s)
4π
(
)
(2.40)
Using the relation between port and diaphragm velocity given in (2.21), this may be re-written in
terms of the diaphragm velocity alone as so
sρ
p1m ( s ) = 0
4π
ρ c2

S p2 S d sV0 c0

 Sd −

sm p + c p + S p2

ρ 0 c02
sVc


 u d (s)


(2.41)
This expression may be written in terms of the input voltage vc(s) using the general impedance form of
the relationship between diaphragm velocity ud(s) and input voltage of (2.27). Separating out the
Chapter 2
48
internal mechanical, front, and rear acoustical parts as per (2.14), the acoustic pressure may be written
in terms of the input voltage as so
ρ c2

S p2 S d sV0 c0
p1m ( s ) sρ 0 
=
Sd −
4π 
vc ( s)
sm p + c p + S p2

ρ 0 c02
sVc

φ0


2
 Z eb ( s ) Z mo ( s ) + Z rmr ( s ) + Z rmf ( s ) + φ 0

(
)
(2.42)
By using the same simplifications as for the closed box analysis presented in §2.1.7, it was shown by
Novak (1959) that the above transfer function can be written as a fourth-order high pass filter. This
description was used to interesting effect by Thiele (1961), who brought the full weight of tools for
active analogue filter design to the problem of design & analysis of loudspeakers in a ported cabinet.
Thiele wrote a table of different response types based upon the different standard alignments for a
fourth-order high pass filter, which gave a compact description of the possible frequency response of a
ported loudspeaker in terms of its physical dimensions and manufacturing specifications.
Specifically, to apply the theory presented by Thiele (1961), these simplifications are made:
Z eb ( s )
≅ Reb
(2.43)
Z mo ( s ) + Z rmf (s) ≅ smt + ct + k d s
Using these simplifications and substituting the expression for the rear radiation impedance presented
in the LHS of (2.23) for Zrmr(s) produces:
ρ c2

S p2 S d sV0 c0
p1m ( s ) sρ 0 
=
Sd −
4π 
v( s )
sm p + c p + S p2

ρ 0 c02
sVc

φ0


2
 Reb (Z rmr ( s ) + smt + c t + k d s ) + φ 0

(2.44)
2.2. Nonlinear models of loudspeakers
Linear models are conceptual simplifications. They are valid for a limited range of operation. For the
electrodynamic loudspeaker, the scope of range is mostly defined by the diaphragm-coil displacement.
The range of applicability of the same model structure as that developed in §2.1 above may be
extended by introducing displacement-dependence on several parameters of the model.
2.2.1. Parametric nonuniformity and causes of nonlinearity
Parametric nonuniformity and other causes of nonlinearity result in nonlinear distortion in the acoustic
signal reproduced by a loudspeaker. Nonlinear distortion reduces the quality of the sound reproduced
by the loudspeaker. This thesis does not aim to determine what degree of nonlinear distortion is
acceptable. Instead, it is assumed that present levels of nonlinear distortion, in existing products, are
proper compromises between construction expense and sufficiently low distortion for market
acceptability.
Causes of nonlinearity in loudspeakers have been thoroughly researched, and are generally thought to
be well-understood.
Identification and modelling of nonlinear mechanisms has been an active academic subject for most of
the last century. The dominant mechanisms may be described as parametric nonuniformity, i.e. the
variation of some linear parameter with excursion by some analytic description. The most common
form for description of parameter non-uniformity is a polynomial power-series expansion in xd, which
Loudspeaker models
49
is used for most models of parameter non-uniformity in this thesis. This description is particularly
convenient for real-time computation, as it requires simple polynomial evaluations in an otherwise
linear model.
Different types of parametric nonuniformity and causes of nonlinearity are discussed in the following
sub-sections.
A. Magnetic field nonuniformity
The magnetic field generated by the loudspeaker’s magnet system, i.e. part (7) shown in Figure 2.1
and in Figure 2.2, will not be completely uniform over the region through which the coil will move.
One may consider a displacement-dependent magnetic field B(xd), which describes the variation of the
magnetic field along the axial direction of the coil’s movement. From this, an effective transduction
coefficient φ(x) may be calculated as so
h
φ( x d ) =
2
∫ B (ξ − x
−h
d
)dξ
(2.45)
2
where h is the height of the voice-coil.
The transduction coefficient may be is approximated a power series expansion of order Nφ
φˆ ( x d ) ≈ φ( x) =
Nφ
∑φ
k
x dk .
(2.46)
k =0
A plot of a typical measurement of the nonuniformity in the transduction coefficient is shown in
Figure 2.11. The data shown in Figure 2.11 is for a 16mm diameter microspeaker of the basic type
shown in Figure 2.1. This data was obtained with a commercial instrument for determining the
coefficients of the polynomial approximation to φ(xd), i.e. φk in (2.46). A description of this
commercial instrument is provided by Klippel GmbH (2001).
φ(x), transduction coefficient (N/A)
0.5
0.4
0.3
0.2
0.1
0
−0.2
−0.1
0
0.1
xd, coil displacement (mm)
0.2
0.3
Figure 2.11: Measured nonuniformity in the transduction coefficient versus diaphragm-coil displacement, i.e.
φ(xd), for loudspeaker type shown in Figure 2.1.
Chapter 2
50
The transduction coefficient φ(xd) is not well-represented by a finite Taylor series if it is evaluated
outside of the range of x-values on which the coefficients were determined. This can be seen in the top
frame of Figure 2.12. This is particularly problematic during simulation of nonlinear behaviour,
where it is interesting to simulate high diaphragm-coil displacement. If the transduction coefficient is
negative, it can result in unstable simulation results.
A solution to this problem is to invert the transduction coefficient variation, and perform a polynomial
fit on this. Representing this as ψ(xd), i.e.
ψ( x d ) =
Nψ
∑ψ
n
n xd
,
(2.47)
n =0
Magnetic Flux Density (Tesla)
the actual transduction coefficient function can be recovered from 1/ψ(xd). A plot of 1/ψ(xd) is shown
in the lower frame of Figure 2.12. This technique is used to ensure stable results in the simulations
presented in §§5.2.1 - 5.2.2.
Polynomial fit, directly on magnetic field data
1
Original data
0 Fit evaluated outside
of range of original data
−1
−5
0 of magnetic field data
Polynomial fit, on inverse
5
Inverse of polynomial fit on0inverse magnetic field data
5
0
Excursion distance (mm)
5
40
20
Magnetic Flux Density (Tesla)
0
−5
1
0
−1
−5
Figure 2.12: Polynomial fitting for force-factor (B•l factor) data; top frame shows how coefficients fit on
direct data, when evaluated outside of range of direct data, can return a negative result. Bottom frame shows
how fitting coefficients on inverse data, and evaluating as inverse function does not produce this problem.
B. Stiffness nonuniformity
Nonuniformity in the suspension exhibits itself either as a smooth, memory-less function of
displacement, as a hysteretic function, or, when the diaphragm-coil assembly contacts other parts of
the loudspeaker, as a discontinuous function. These three different types of stiffness nonuniformity
are discussed separately below.
Gradual (‘smooth’) variation in the suspension
It is known that the stiffness presented by the suspension will not be uniform throughout the range of
displacement. Typically, the stiffness will increase with higher displacements. This is known in
mechanics as a ‘hardening spring.’
Loudspeaker models
51
This effect was analysed in some detail by Olson (1944), illustrating what is now referred to as the
‘jumping’ phenomenon. This refers to sudden changes in the response amplitude as the frequency of a
sinusoidal excitation is swept upwards or downwards around the resonance frequency. Due to the
fact that, with a hardening spring, the resonance frequency increases with increasing displacement, for
a certain range of frequencies just above the small-signal resonance frequency, there will not be a
unique response amplitude solution at higher levels. As the frequency is changed past a certain critical
point, the response amplitude will ‘jump’ from the amplitude of one stable response to another.
The stiffness may be approximately described by a truncated power series, forming an order Nk–th
order polynomial as so:
k d ( xd ) =
Nk
∑k
n
n xd
(2.48)
n =0
A plot of the nonuniformity of the stiffness on a typical microspeaker is shown in Figure 2.13.
4500
3500
3000
2500
2000
1500
d
k , suspension stiffness (N/m)
4000
1000
500
0
−0.2
−0.1
0
0.1
xd, coil displacement (mm)
0.2
0.3
Figure 2.13: Measured nonuniformity in the suspension stiffness with respect to diaphragm-coil displacement,
i.e. kd(xd), on a sample of the loudspeaker type shown in Figure 2.1.
As can be seen in Figure 2.13, the stiffness variation does not appear as a ‘hardening spring,’ wherein
the stiffness would increase with the absolute value of the displacement. This ‘hardening spring’
phenomenon is the most common form of stiffness nonuniformity seen in typical loudspeakers.
Instead, it is clear from Figure 2.13 that the suspension of the microspeaker produces a stiffness that
increases with forward displacement and decreases with rearward displacement.
Stiffness
nonuniformity of this shape is thought to be characteristic of the microspeaker with a suspension of the
shape shown in Figure 2.1.
Buckling
Through various experiments and measurements made for this thesis, it was found that in some cases
the suspension stiffness may have exhibit some buckling, producing strong hysteresis (backlash). A
measurement on a loudspeaker with such an effect is shown in Figure 2.14. This kind of stiffness
nonuniformity cannot be modelled with a function of the type in (2.48). A more complicated, memorydependent model would be needed. As this type of stiffness is a manufacturing defect to be avoided
by proper mechanical design, a model of this effect is not considered in this thesis.
52
Chapter 2
hysteretic effect as diaphragm
moves 0.2mm towards magnet
Figure 2.14: X-Y Voice-coil current–displacement plot for a microspeaker, showing effect of buckling in the
suspension, at how it leads to ‘backlash’ in the displacement/current X-Y plot. Displacement is measured with
a laser displacement interferometer (fringe counter), in an otherwise similar arrangement to that shown in
Appendix A.
Suspension limit
There is a physical limit to the distance that the diaphragm-coil assembly may move. As the
diaphragm moves forward, away from the magnet, the stiffness presented by the suspension will
gradually increase, until the suspension is no longer flexing, but is instead under tension.
As the diaphragm moves rearward, the coil and/or the diaphragm will at some point contact the
magnet and/or frame. At this point, the suspension stiffness kd becomes ‘infinite,’ in the sense that no
matter how much force is applied to the coil, the diaphragm-coil assembly will not move any further.
This effect is referred to be some as ‘bottoming-out’ of the suspension, by analogy to a similar effect
occurring in automobile suspensions.
The maximum displacement set by this suspension limit for the microspeakers investigated in this
thesis is approximately 0.35mm peak.
This limit to the suspension can be modelled by setting the suspension to be infinite at the limiting
value. The suspension is not modelled in this way in other parts of this thesis. Particularly, it is not
modelled in the context of compensation of nonlinear distortion. Were this to be modelled, an attempt
to compensate for the suspension limit would create an infinite gain in the compensator, which, to be
sure, would be somewhat impractical.
C. Inductance nonuniformity
The electrical inductance Leb will vary with coil position. As the coil moves forward, away from the
magnet system, there will be less ferromagnetic material ‘seen’ by the magnetic field generated by the
coil, producing a lower inductance. An example of this type of variation is shown in Figure 2.15. As
can be seen in this figure, the inductace decreases with forward movement of the voice-coil, wherein
there is less ferromagnetic material seen by the voice-coil.
Loudspeaker models
53
90
Leb, blocked electrical inductance (µH)
80
70
60
50
40
30
20
10
0
−0.2
−0.1
0
0.1
xd, coil displacement (mm)
0.2
0.3
Figure 2.15: Measured nonuniformity in the blocked electrical inductance with respect to diaphragm-coil
displacement, i.e. Leb(xd), for the loudspeaker type shown in Figure 2.1.
As shown in Figure 2.5, the blocked electrical inductance makes a negligible contribution to the total
electrical impedance in the speech-frequency range for the microspeaker. The variation of this
inductance, though measurable, does not affect its overall performance at large signals. The
nonlinearity produced by inductance nonuniformity is, therefore, only considered in the discussion on
theory of nonlinear simulation presented in §2.2.2, and not in other parts of this thesis.
D. Mass nonuniformity
It was shown by Olsen and Thorborg (1995) that the effective mass of the loudspeaker diaphragm can
change with position. It was found that the roll surround used in many loudspeakers produces a
changing mass depending on whether the diaphragm is near its forward or rearward displacement
extremity. This produced an effective moving mass which changed by some ±20% from its value at
the equilibrium (rest) position. The variation of the mass with displacement was found to be wellmodelled by a first order polynomial expansion in xd.
Over the limited range of displacements of which the microspeaker considered in this thesis is capable,
it is assumed that the mass is uniform.
E. Area nonuniformity
In the same study by Olsen and Thorborg (1995) that discovered mass-variation, it was also found that
the effective area changes with displacement. This was found to be due to the same mechanism that
caused the mass-variation, the changing behaviour of the surround. As the surround (outer
suspension) is extended rearward from equilibrium, it contributes more to the effective radiation area.
As it is extended forward from equilibrium, it contributes less. Over the range of possible excursions,
variations in the effective area were found to be on the order of ±12%. Similar to the mass variation,
the area variation was found to be well-modelled by a first-order polynomial in xd.
Over the limited range of displacements of which the microspeaker considered in this thesis is capable,
it is assumed that the effective area is uniform.
Chapter 2
54
F. Magnetic attraction force
The magnetic attraction force as it appears in electrodynamic loudspeakers was first discussed by
Cunningham (1949). This effect is the result of the classical electrodynamic effect of the attractive
force existing in a current-carrying wire for any ferromagnetic material in its vicinity. For the case of
an electrodynamic loudspeaker, the coil will exhibit this force for the material in the magnet and
magnetic circuit. It was noted by Cunningham that this force is related to the spatial derivative to the
internal inductance of the coil, i.e.
Fma =
1 2 dLeb (xd (t ) )
ic (t )
2
dxd (t )
(2.49)
This ‘solenoid’ effect is problematic in largely-overhung voice-coils, which will produce significant
gradients on the electrical inductance Leb with respect to displacement of the coil. It was noted by
Cunningham that this effect is reduced if the ferromagnetic material in the magnet system is near or at
magnetic saturation.
This phenomenon introduces an additional forcing term to the force equation. As the force varies with
the product of the square of the current ic(t) and derivative of the inductance, it will render the
resulting differential equation nonlinear.
This effect is dependent upon nonuniformity in the electrical inductance. As explained in part C,
above, nonuniformity in the electrical inductance is not considered for the speakers considered in this
thesis. Thus the nonlinear mechanism of the magnetic attraction force is not considered in this thesis.
G. Frequency modulation distortion
Frequency modulation distortion results from the fact that the diaphragm will have a finite velocity as
it vibrates. This effect has seen considerable academic interest (Klipsch, 1968), (Butterweck, 1989).
Simulations of the magnitude of nonlinear distortion produced by this effect show that it is several
orders of magnitude below that created by nonuniformity in the transduction coefficient and
suspension stiffness (Zóltogórski, 1993).
2.2.2. Nonlinear simulation
From this point forward, only models of parametric nonuniformity which operate as zero-memory
nonlinear systems will be considered. By ‘zero-memory,’ it is meant here that the model of
nonlinearity depends only on instantaneous quantities of the system. This is a deliberate choice, as it
makes analysis of the resulting nonlinear system significantly more simple than if the models were to
conclude some ‘memory,’ i.e. to depend on past quantities of the system. Not all types of parametric
nonuniformity can be modelled in the this way. For example, the hysteretic stiffness shown in Figure
2.14 above does depend on past values of the displacement. This type of effect is not modelled in this
thesis. This is because the end purpose of the nonlinear modelling that follows is to form the basis of
an electronic processor for compensation of nonlinearity. It is judged that such memory-dependent
parametric nonuniformity cannot be accurately modelled with practical controllers. It is furthermore
judged that these types of problems must be solved by proper mechanical design, as they generally can
not – in practice – be compensated for electronically.
Loudspeaker models
55
Introducing the parametric nonuniformity discussed above into the voltage equation of (2.1) produces
vc (t ) = Rebic (t ) + Leb (xd (t ) )
dic (t )
dx (t )
dL (x (t ) ) dxd (t )
+ φ(xd (t ) ) d + ic (t ) eb d
dt
dt
dx
dt
(2.50)
The parametric nonuniformity may be introduced into the force equation in a similar manner. Using
(2.12) to substitute the current for the force as the forcing term, and adding the magnetic attraction
(‘solenoid’) force term to the RHS, the force equation becomes:
ic (t )φ( xd (t ) ) = mt (xd (t ) )
d 2 xd (t )
dx (t )
dL ( x (t ) )
+ ct d + kt (xd (t ) )xd (t ) − 12 ic2 (t ) eb d
2
dt
dxd (t )
dt
(2.51)
Solution of these equations may be used to simulate the loudspeaker’s behaviour given appropriate
descriptions of the parameter nonuniformity. The equations are nonlinear in the displacement, xd(t),
and therefore a general frequency-domain solution for the displacement in terms of the input voltage is
not possible, as it is for the linear case presented in §2.1.6 above.
The Volterra series is a natural method for developing a general solution to the nonlinear equations. It
theoretically permits precisely the same type of general solution for xd(s) in terms of the input vc(s) as
is developed for the linear case above, in a quasi-frequency domain manner. Its application to the
loudspeaker was first presented by Kaizer (1987). The Volterra series becomes highly complicated
when higher than second-order nonlinearity (parameters which depend on a power of xd greater than 2)
are considered. Therefore although it has been demonstrated in literature that this method can be used
to measure and compensate for loudspeaker nonlinearities with some success, it makes a poor general
method for nonlinear analysis.
A more simple strategy for solving nonlinear differential equations than the Volterra series is
Harmonic Balance. In this method, the system is analysed in the frequency domain, for a finite
number of excitation frequencies. The frequency components generated in other terms in the system
are iteratively calculated until the resulting error is reduced to some desired value. This method was
proven effective by Klippel (June 1992) for single and multi-frequency excitation, for predicting
harmonic and intermodulation distortion, as well as the ‘jumping effects’ found by Olson, described in
§2.2.1, part B above. However, it is not suitable for computation on more complex waveforms or long
arbitrary time series, whose frequency-domain description would require large numbers of frequency
components in the input signal.
NARMAX modelling (Nonlinear auto-regressive moving-average with exogenous input) has been
proposed as a method for identification of nonlinearities (Jang and Kim, 1994), and for nonlinear
simulation (Potirakis et al., 1999). However, it is not suitable for developing an input-output
characterisation from such nonlinear differential equations as Eqs. (2.50) and (2.51), without first
deriving sample input-output time series by some other method. This is because there has yet to be
any method established for determining coefficients of a NARMAX model from those of a nonlinear
differential equation. All published work on NARMAX modelling to date is based on adaptation of the
model to measured input-output time series data, and not determination from physical constants of a
system. Thus NARMAX modelling does not permit simulation in the sense of the prediction of the
performance of system based on its design characteristics without first building the system.
Furthermore, it does not permit the development of a connection between the amount of nonlinearity
of a loudspeaker to its design characteristics. The main utility in the NARMAX model is the ability to
predict the nonlinear response of the loudspeaker for some arbitrary signal.
Chapter 2
56
A common method for solving differential equations is numerical integration. Numerical integration
was used for auralisation of nonlinearity by Christensen and Olsen (1996). It has also been used to
predict harmonic and intermodulation distortion. This method is used in chapter 5, below, to analyse
the changes in transduction coefficient nonuniformity, φ(xd), caused by shortening of the voice-coil
height.
2.3. Discrete-time physical modelling
In the loudspeaker models developed in §§2.1-2.2, all time-dependent quantities were defined with
respect to continuous-time. In contrast, textbooks and other literature on modern digital signal
processing theory and applications invariably consider controllers and processors operating in
discrete-time. Practical implementation of active control can only be implemented by a digital
processor.1 The adaptive controller will need a model of the loudspeaker in order to perform system
identification, for updating the feedforward part of the controller. This is not straightforward, as the
basic parametric descriptions of systems in continuous-time is not the same as that in discrete-time.
This is a basic problem for discrete-time control systems dealing with real-world plants, and will be
considered at a fundamental level in this section.
This problem may be described succinctly as follows. Real, physical systems, are generally described
by systems of differential equations, defined by two coefficient vectors c and d. These may be
represented, along with the respective Laplace-domain transfer-function, as so:
cN
d ( N ) y (t )
dy (t )
d ( M ) x(t )
dx(t )
c
c
y
t
d
+
...
+
+
(
)
=
+ ... + d1
+ d 0 x(t )
1
0
M
(M )
(N)
dt
dt
dt
dt
M
∑d s
n
⇒ H (s) =
n=0
N
∑c s
n
n
(2.52)
n
n =0
Controllers handle systems with digital processing operating in discrete-time, defined by two
coefficient vectors a and b. These generally describe systems in terms of difference equations,
conveniently analysed by their z-domain transfer function:
a N y[ k − N ] + ... + a1 y[k − 1] + a 0 y[k ] = b M x[k − M ] + ... + b1 x[k − 1] + b0 x[k ]
M
⇒
H ( z) =
∑b
nz
−n
∑a
nz
−n
n=0
N
(2.53)
n=0
Methods do exist for determining a set discrete-time coefficients a and b, directly from c and d, giving
a discrete-time system with approximately the same frequency response as the continuous-time system
described by c and d. However, these typically require detailed calculation, and suffer from numerical
sensitivity complications, as is described below. From elementary discrete-time signal processing
theory (e.g. Oppenheim and Schafer, 1989), there are two formal methods for determining the
coefficient vectors c and d from b and a: the bilinear transformation, and impulse invariance.
1
It is theoretically possible to implement such controllers with analogue processors. However, they are
generally subject to drift, are difficult to program, and cannot be easily controlled by microcontroller (as would
be needed for an actual commercial product). For these reasons, analogue implementation of an adaptive
feedforward controller for a loudspeaker is considered impractical.
Loudspeaker models
57
The bilinear transform determines a z-domain transfer function by the substitution
s = 2 Fs
1 − z −1
(2.54)
1 + z −1
It can be shown that the discrete-time coefficient vectors a and b resulting from the bilinear
transformation are determined by matrix multiplication between the continuous-time coefficients c and
d and matrices P and Q as so
a = Pc
b = Qd
(2.55)
where the columns of P and Q are polynomial coefficients resulting from, respectively, the M th and
N th binomial expansion of the products between (1 – z –1) and (1 + z –1). The coefficient vectors c and
d from the differential equation contain the physical information of the plant, and are thus those
parameters of interest to the feedforward controller. They may be determined from the discrete-time
coefficients a and b by multiplication by P-1 and Q-1 respectively. Although the matrices P and Q are
generally invertible, they are fully populated, making this multiplication somewhat complicated –
complicated to the extent that it would be unsuitable for real-time processing.
The method of impulse invariance performs a discrete-time sampling of the impulse response of the
continuous-time system. The use of this method thus requires conversion of the polynomial ratio of
(2.52) in s to a partial fraction expansion, from which the impulse response in continuous-time may be
obtained by inverse Laplace transform. A discrete sampling of this continuous-time impulse response
may be z-transformed, producing
H ( z) =
1
Fs
N
∑1− e
n =1
An
λ n Ts
z −1
(2.56)
where An are the numerator coefficients of the partial fraction expansion, and λn are the roots of the
continuous-time denominator polynomial of (2.52). The expression in (2.56) may be expanded into
the standard polynomial ratio in z-1 of (2.53), resulting in a set a of coefficients suitable for a digital
filter. The net effect of the impulse invariance method is to map the poles from the s-plane to the zplane according to
π n = e Ts λ n
(2.57)
where πn is the nth pole in the z-plane, and Ts is the sampling period. The sampling of this impulse
response produces familiar aliasing effects. These aliasing effects can usually be compensated for by
matching the overall-gain between the continuous- and discrete-time frequency response functions at
some frequency for which the continuous-time frequency response function is simply defined.
However, perhaps more problematic, is that this method requires that the roots of the continuous-time
denominator polynomial be calculated. Such a calculation is difficult, and can fail due to finiteprecision effects even for floating-point computation. Thus this method, in its formal form, is also
considered unsuitable for development of a discrete-time model.
Due to these problems, specific discrete-time models of the loudspeaker are developed below. The
exponential pole-mapping of (2.57) is used as a starting point, with slight modifications the ensure the
frequency response of the discrete-time models matches that of the continuous-time. Particular effort
has been made to ensure that the parameters of these models have direct physical interpretation.
Chapter 2
58
2.3.1. FIR filter for electrical admittance
As discussed in the introduction, an adaptive feedforward controller requires an adaptive filter for
system identification. The most well known and reliable methods for adaptive filtering use FIR filters
for plant models. This is due to the fact that FIR filters are inherently stable, and that their error
surfaces for any system are unimodal, without local minima. For this reason, an FIR filter describing
the electrical admittance of the loudspeaker is investigated. The electrical admittance of the
loudspeaker is considered because, as explained in the introduction, the electrical current is the only
practical feedback signal available from the loudspeaker.
The suitability of an FIR filter for identifying a loudspeaker’s parameters by its electrical impedance
may be determined by analysis of the impulse response of the electrical admittance. As per the
formal method of impulse invariance described above, an analytical expression for the impulse
response may be obtained by inverse Laplace transform of the Laplace-domain expression for the
transfer function. To consider how a digital FIR filter would react to the electrical admittance of a
loudspeaker, distinct points of the resulting continuous-time impulse response are analysed. This is
somewhat different to the formal impulse invariance method, wherein the individual terms of the
analytical expression for the impulse response are converted to the z-domain (using special properties
of the z-transform), and re-combined to the single ratio of powers in z-1 of (2.53). This method is not
used her , as it would result in an IIR filter, whereas the original stated purpose here was to develop an
FIR filter.
For the simplest case of a loudspeaker mounted in a closed box, this impulse response may be
determined by inverse Laplace transform of the multiplicative inverse of the s-domain expression for
the total electrical impedance given in (2.39). This may be expressed as so:
Ae ( s ) =
1
1 ω 02
−
m t Ym · e ( s )
Reb ω 0 Qes Reb
(2.58)
Amp/Volt (Mag.)
where Ym·e(s) is the mechanical mobility as described by (2.6) in §2.1.2, with an important difference.
The term Ym·e(s) in (2.58) includes the effective damping produced by the ‘back EMF.’ In other words,
the damping in Ym·e(s) will be given by the sum of the mechanical damping ct and the ‘electrical
damping, φ02 Reb . A plot of the magnitude and phase of the frequency response of (2.58) is shown in
Figure 2.16.
0.1
0.05
0 2
10
3
10
4
10
Phase
pi/2
0
−pi/2 2
10
3
10
Frequency Hz
4
10
Figure 2.16: Magnitude and phase of the electrical admittance transfer function in (2.58) for a typical set
parameters.
Loudspeaker models
59
It may be shown that the inverse Laplace transform of (2.58) is given by
h Ae (t ) =

ω ζ
1
1 ω 02 −ω0ζt 
 cos ω d t − 0 sin ω d t 
δ(t ) −
e
ω 0 Qes Reb
ωd
Reb


(2.59)
The discrete-time representation of (2.59) is not given simply by the periodic sampling of the
continuous-time time function as is done in the impulse invariance method described above, or for that
matter in digital data acquisition. Instead, it is necessary to integrate over each sample period n, i.e.
Ts 2 + nTs
hAe [ n] =
∫h
Ae (t ) dt
(2.60)
−Ts 2 + nTs
In this way the delta function in (2.59), δ(t) / Reb, becomes an impulse of height 1 / Reb. A plot of this
discrete-time impulse response for a typical set of loudspeaker parameters is shown in Figure 2.17.
Amp / Volt
1/Reb−A/2
Damping Decay
0
−A
0
T0/2
T0
3T0/2
2T0
Time (s)
5T0/2
3T0
7T0/2
Figure 2.17: Impulse response of (2.59) for same set of parameters as Figure 2.16. Points on the impulse
response denote locations of FIR filter taps as would occur given Fs = 8kHz. The x-axis tick marks are shown
at intervals of T0, which is 1/f0, where f0 is the resonance frequency.
An adaptive FIR filter operating on a loudspeaker’s electrical admittance will identify discrete points of
(2.59), as shown in Figure 2.17. From sufficient number of these points, the parameters may be
identified by minimising the error between the values of these points and the formula of (2.59).
This method is considered too computationally expensive. This is because two computationally
intense steps must be performed:
• Adaptation of sufficient length FIR filter
• Determination of parameters from identified FIR coefficients.
Figure 2.17 shows one of the best-case situations with respect to number of taps needed in the FIR
filter. In this case, the ratio of the resonance frequency f0 to the sampling frequency Fs is about 0.1.
Here it is clear that at least 15 taps would be needed in order to make an accurate estimate of the
resonance frequency and damping factor.
Chapter 2
60
Perhaps even more difficult, however, is the final determination of the parameters ω0, ζ and φ0 from
the FIR filter coefficients. The error function resulting between these coefficients and (2.59) would be
highly nonlinear in the parameters ω0, ζ and φ0, and would thus have to be determined by an iterative
method (e.g. Newton’s method). This type of meta-calculation is considered, for the present work,
unsuitable for real-time processing.
2.3.2. IIR filter for receptance of an SDOF system
Knudsen et al. (1989) presented a formula for a second-order IIR filter giving the frequency response
of a second-order mechanical system, and used this as a model of a loudspeaker’s mechanical
dynamics. In their paper, it is explained that the parameters of the IIR filter are related to the physical
parameters of the mechanical system by simple expressions not presented in the paper. As these
expressions will be used in several parts of this thesis in some detail, their derivation is presented here.
The s-domain transfer function for the mechanical receptance of a single mass-spring-damper (SDOF)
system, according to elementary mechanical dynamics (e.g. Newland, 1989), is given by
X m (s) =
1
xd ( s )
=
2
f c ( s ) mt s + ct s + kt
(2.61)
In a method similar to that in §2.1.2, above, this may be written in factored, form,
X m ( s) =
1
1
m t ( s − λ 1 )( s − λ 2 )
λ 1 , λ 2 = − ω 0 ζ ± iω 0 1 − ζ 2
(2.62)
The terms λ1 and λ2 are the poles of the transfer function, as they are the roots of the denominator
polynomial in (2.61). As per impulse invariance described above, the basic form of a discrete-time
transfer function may be obtained by mapping the poles of (2.62) to the z-plane, according to
π n = e Ts λ n . This provides
H X m ( z) = σx
z −1
1 + (− π1 − π2 ) z −1 + π1π2 z − 2
(2.63)
where σx is the characteristic sensitivity in z-domain, which shall be defined below. This z-domain
representation may be written in standard form as
H X m ( z) =
b1 z −1
1 + a1 z −1 + a 2 z − 2
.
(2.64)
where
b1 = σ x
a1 = −(π1 + π 2 )
(2.65)
a 2 = π1 π 2
In this way the coefficients of the discrete-time transfer function are determined from those of the
continuous-time version according to:
a1 = −e − ω z ζ+ iω z
1− ζ 2
− e − ω z ζ− iω z
= −2e − ω z ζ cos  ωz 1 − ζ 2 


a2 = e − 2 ω z ζ
1− ζ 2
(2.66)
Loudspeaker models
61
The z-domain characteristic sensitivity, σx, is determined by matching at a certain frequency the value
of the z-domain frequency response in (2.63) to the s-domain frequency response in (2.61). The
characteristic frequency is one for which the transfer function is most simply defined. For the
mechanical receptance, this occurs at f = 0 (i.e. 0 Hz), where Xm(s) in (2.61) is 1/kt. With the
foregoing, σx is
σx =
1 + a1 + a2
.
kt
(2.67)
The frequency response and pole-zero map of this discrete-time model of the receptance is shown in
Figure 2.18, for three different resonance frequencies, and for three different damping values. The
frequency response of the discrete-time model of receptance closely matches the continuous-time
response in both magnitude and phase. The only difference occurs at high frequencies, close to the
Nyquist frequency. At these high frequencies, the discrete-time model has a higher magnitude
response than the continuous-time response. Due to the low-pass nature of the receptance function, it
is assumed that the total percentage error of a broad-band signal filtered by this discrete-time model of
the receptance would be minimal. If better accuracy were needed up the Nyquist frequency, this could
be achieved by increasing the complexity of the filter, i.e. by increasing its order.
Broken: s−domain
Solid: z−domain approximmation
0.8
0.6
−20
0.4
−40
100
1000
Fs/2
Phase (radians)
0
Imaginary Part
dB re 1mm/N
1
0
0.2
0
−0.2
−0.4
−0.6
−pi/2
−0.8
−1
−pi
100
1000
Frequency (Hz)
Fs/2
−1
−0.5
0
0.5
1
Real Part
Figure 2.18, [left]: Frequency response comparison between continuous-time (broken) and discrete-time
(solid) representation of the receptance; [right]: pole-zero map in z-plane of receptance model.
The continuous-time resonance frequency ω0 and the damping ratio ζ can be determined from a1 and
a2, given certain limits on the range of values of a1 and a2. In the s-plane, the damped natural
frequency is the distance of the poles, λ1 and λ 2 in (2.62), along the Imaginary axis. In other words,
the damped natural frequency is given by the imaginary part of the poles, λ 1 and λ 2. By the analogy
that the imaginary axis of the s-plane is wrapped around the unit circle of the z-plane, the same
damped natural frequency, normalised to the sampling frequency Fs, in z-domain will be given by the
angle of the pole around the unit circle of the z-plane. Therefore the damped natural frequency is
given by
 Im{π1} 

ω0Ts 1 − ζ 2 = arctan
 Re{π1} 
(2.68)
where π1 is the value of the pole in the z-plane. The real and imaginary parts of the pole π1 are given
by relating the factored from of (2.64) to its polar form in (2.63), resulting in
Chapter 2
62
Im{π1} =
− a12 + 4a2
1
2
for − a12 < 4a2
Re{π1} = − 12 a1
(2.69)
A clue to finding the damping ratio ζ from a1 and a2 lies in recalling that it is related to the pole’s
distance to the imaginary axis in the s-plane, which in the z-plane translates to its distance to the unit
circle. This distance is given by 1 − a2 . Given the definitions of a2 in (2.66), the square-root term
is
(
)
a 2 = e − ωs ζ .
(2.70)
Therefore given the condition a2 > 0 we have
( )
− ωz ζ = ln a2
(2.71)
Inspection of Eqs. (2.68) and (2.71) shows the s-plane pole can be reconstructed from a1 and a2 as so

 Im{π1} 

λ1 , λ 2 = Fs ln a2 ± i arctan

 Re{π1} 
( )
(2.72)
Recalling that the undamped natural frequency can be determined from the magnitude of the
continuous-time eigenvalue, i.e. ω0 = λ1 , the undamped natural frequency may be determined from
a1 and a2 as so
ωz = ln 2

( a ) + arctan  −
2
2

− a12 + 4a2 

a1

(2.73)
= ω0 Fs
The damping ratio may be obtained from the above result and (2.71), giving:
ζ=
( )
− ln a2
ω0Ts
(2.74)
2.3.3. IIR filter for mobility of an SDOF system
As discussed in §2.1.2, the continuous-time transfer function of the mechanical mobility of an SDOF
system is given in s-domain by
Ym ( s ) =
s
s mt + sct + kt
2
(2.75)
A discrete-time approximation to the continuous-time description by using the same process as used
for the mechanical receptance described in §2.3.2.
Thus we have
Ym ( z ) =
σu − σu z −2
1 + a1 z −1 + a2 z − 2
(2.76)
where a1 and a2 are as defined in (2.65). The reference sensitivity σu is defined by matching the
overall value of the discrete-time frequency response function to that of the continuous-time frequency
response function. The continuous-time frequency response function for the mechanical mobility is
most simply defined at f = ω0, where Ym(s) = 1/ct. Therefore the σ0 is defined by matching the two
frequency response functions at this frequency, providing
Loudspeaker models
σu =
63
1 1 − ( π1 + π2 )e −iω0Ts + π1π2e −2iω0Ts
.
ct
1 − e − 2iω0Ts
(2.77)
The frequency response and pole-zero map of this discrete-time model of the mobility is shown in
Figure 2.19. The response of the discrete-time model of mobility closely matches the continuous-time
response in both magnitude and phase. Like the receptance function, the only difference occurs at
high frequencies, close to the Nyquist frequency. Here, the discrete-time model has a lower
magnitude response than the continuous-time response. The agreement between these two is
considered acceptable for the present application.
1
10
0.8
0
0.6
−10
0.4
−20
100
1000
Fs/2
pi
Imaginary Part
dB re 1.0 (m/s)/N
Solid: z−domain Broken: z−domain
20
0.2
0
−0.2
Radians
−0.4
−0.6
0
−0.8
−1
−pi
100
1000
Frequency (Hz)
Fs/2
−1
−0.5
0
Real Part
0.5
1
Figure 2.19 (a) – (c): Frequency response comparison between continuous-time (broken) and discrete-time
(solid) representation of the mobility; [right]: pole-zero map in z-plane of mobility model.
The reference sensitivity σu can be defined in terms of a1 and a2 if the total mass mt is fixed. By using
the substitution ct = 2ζω0 mt we have
σu =
1
1 + a1e − iω z + a2e −2iω z
2ζω z Fs mt
1 − e − 2iω z
(2.78)
The terms ζ and ωs may be determined from a1 and a2 as described in §2.3.2 above.
For real-time, real-valued processing, the definition of σu in (2.78) is too complicated. It has been
found that σu depends only on the term a2. Furthermore, it has been found that, over the most
interesting range of values of a2, σu may be determined approximately by a polynomial approximation
as so
σu =
1
mt Fs
N
∑p
n=0
n
σ u · n a2
(2.79)
where p σu ·n is the nth coefficient of a polynomial approximation to the function of σ0 in terms of a2.
As shown in Figure 2.20, a third-order polynomial approximation (N =3) provides a good compromise
between accuracy and complexity. An example of the accuracy of this polynomial fit is shown in
Figure 2.20. Although there is significant deviation for a2 below 0.4, only the range of a2 from 0.6 to
1.0 is of interested for the type of loudspeaker of interest in this thesis. This can be seen clearly in
Figure 4.9, in the discussion on the ‘tolerance quadrilateral,’ in §4.1.5.
Chapter 2
64
σ , 3rd order polynomial apprxomation; Broken: original, Solid:Polyfit
u
0.5
0.4
t
σ , normalised to F m =1
0.45
s
0.35
0.3
u
0.25
0.2
0.15
0.1
0.05
0
0.2
0.4
0.6
0.8
1
a2
Figure 2.20: Reference sensitivity, σu vs. a2; actual value (broken) vs. 3rd-order polynomial approximation
(solid). Only the range of a2 from 0.6 to 1.0 is of interest, and thus the deviation below 0.4 is not of concern.
2.3.4. Nonlinear discrete-time loudspeaker model
A linear model for the loudspeaker can be developed in discrete-time in the same manner as it is for
the mechanical receptance and mobility in the preceding two sections. This is possible because, for
the case of the loudspeaker mounted in a closed box, the mechanical dynamics of the system have a
single degree of freedom. Accurate representation of the frequency response of single-degree-offreedom mechanical dynamics is possible in discrete-time using the IIR filters for receptance and
mobility described in §2.3.2 and 2.3.3, respectively, above. As mentioned above, Knudsen at al.
(1989) used this discrete-time representation for parameter determination by digital system
identification. This linear representation is used here. The linear representation is combined with
nonlinear components, to produce a discrete-time nonlinear model. Using a technique developed by
Klippel (1992), all nonlinear components are described by zero-memory systems. The key advantage
of this method developed by Klippel is that the linear and nonlinear components are kept separate.
Only the dominant causes of nonlinearity are included in the nonlinear model presented here.
Specifically, these are nonuniformity in the transduction coefficient φ(xd) and suspension stiffness
k(xd) .
In §2.1.2, the mechanical dynamics of the loudspeaker are presented in terms of the mechanical
mobility of an SDOF system. One can equally consider the mechanical receptance, wherein the
displacement is considered as the output (instead of the velocity, for the mobility). Beginning with the
linear discrete-time filter for the receptance derived in §2.3.2 above, the diaphragm displacement is
given in terms of the force applied on the coil by the following difference equation:
xd [ n] = σ x f c [n − 1] − a1 xd [n − 1] − a2 xd [ n − 2]
(2.80)
The force on the voice-coil fc[n] may be calculated from the voice-coil current by
f c [n] = φ(xd [n])ic [n] − k1 (xd [n]) xd [n]
(2.81)
Loudspeaker models
65
where k1(xd[n]) is the variation of the suspension stiffness with displacement, excluding its value at
equilibrium (kd). Thus, if we consider the stiffness represented by the polynomial expansion of (2.48)
in §2.2.1 part B, k1(xd[n]) is given by
k1 (xd [n]) =
Nk
∑ k x [ n]
l
l d
(2.82)
l =1
By dropping the electrical inductance term, the voice-coil current may be determined using the voltage
equation of (2.50) as so1
ic [n] =
1
[vc [n] − φ(xd [n])ud [n]]
Reb
(2.83)
The velocity ud[n] appearing in (2.83) may be determined by differentiating the displacement signal.
This differentiation may be represented by the following difference equation
ud [ n] = hdt [ n] ∗ xd [n]
= bdt ·0 xd [n] + bdt ·1 xd [n − 1] − adt ·1ud [n − 1]
(2.84)
In principle values of the filter coefficients in (2.84) may be taken from the bilinear transform,
discussed in the introduction to this sub-section, above. This will, however, lead to an unstable filter,
and thus it is necessary to use a slightly modified value of adt·1. This is discussed in more detail below.
An explicit difference equation for calculating the displacement from the input voltage may be derived
by combining (2.80) - (2.84), as so:


 1

{vc [n − 1] − φ(xd [n − 1])hdt [n] ∗ xd [n − 1]} − k1 (xd [n − 1]) xd [n − 1]
xd [ n] = σ x φ(xd [n − 1]) 

 (2.85)
 Reb

− a1 xd [n − 1] − a2 xd [n − 2]
A diagram showing the flow of processing of (2.85) is shown in Figure 2.21. An extremely important
feature of (2.85) is that it is an explicit equation for predicting the displacement xd[n] from the input
voltage vc[n], incorporating nonlinear models of the transduction coefficient and suspension stiffness.
This is to say that only delayed samples of xd[n] appear in the RHS of this difference equation.
Furthermore, the input signal, the voltage vc[n] also appears only as a delayed sample in the RHS.
Equation (2.85) may, therefore, be seen as a one-step predictor for the diaphragm displacement from
the input voltage. This important feature will be exploited for the development of a method for
compensating nonlinear distortion produced by nonuniformity of the transduction coefficient and
suspension stiffness, presented in §3.3.7.
1
Although (2.50) is expressed in continuous-time, it contains no differential or other time-dependent operators,
and may, therefore, be directly converted to discrete-time.
Chapter 2
66
vc[n]
φ(x)
+
Σ
1/Reb
ic[n]
+
-
Σ
fc[n]
-
Linear mechanical
model
xd[n]
k1(x)
φ (x)
bdt·0
ud[n]
Σ
Σ
z-1
bdt·1 adt·1
Figure 2.21: Flow diagram for the nonlinear prediction of displacement according to (2.85).
One problem with (2.85) is that it is inherently unstable, due to the nature of the approximation of
differentiation made by the bilinear transformation used to derive the velocity signal by hdt [n] ∗ x d [n] .
This velocity signal, when fed back to the input, leads to an instability. This can be seen in the zdomain transfer function of (2.85) valid for the small-signal (linear) case. This z-domain transfer
function is given by
H X m ( z) =
(
(σ x φ 0 Reb ) z −1 + (σ x φ 0 Reb ) z −2
)
(
)
1 + 2 Fs σ x φ 02 Reb + a dt + a1 z −1 + − 2 Fs σ x φ 02 Reb + a 2 + a dt + a1 z − 2 + a dt a 2 z −3
(2.86)
and a pole-zero plot for a typical set of parameters and for adt = 1 is shown in Figure 2.22. As can be
seen on the left-hand side of the pole-zero plot in Figure 2.22, there is a pole (indicated by an ‘x’)
outside of the unit circle – the basic characteristic for instability in an IIR filter. This problem will lead
to high-frequency oscillation, rendering the filter useless. The problem can be corrected by reducing
the value of adt to less than unity (<1.) The pole-zero and frequency response of the same formula but
with adt = 0.85 are shown in Figure 2.23. Here, it is shown that reducing adt in this way moves the
pole inside the unit circle, thereby insuring stability, without significantly affecting the magnitude or
phase response of the filter in the pass-band.
Loudspeaker models
67
Solid:Hxmvc(z) Broken: Hxmvc(s)
Pole−zero plot of Xd(z) / Vc(z); w0 / ws = 0.07
1
−30
0.8
−40
0.6
−50
0.4
−60
−70
100
1000
Fs/2
Phase (radians)
0
Imaginary Part
dB re 1.0 mm/Volt
−20
0.2
0
−0.2
−0.4
−0.6
−pi/2
−0.8
−1
−pi
100
1000
Frequency (Hz)
Fs/2
−1
−0.5
0
0.5
1
Real Part
Figure 2.22: Linear analysis of Eq. (2.85), for adt·1=1; left: frequency response, magnitude (upper) and phase
(lower); right: poles and zeros in z-plane. Notice pole (indicated by an ‘×’) just outside of the unit circle on the
left-side of the plane.
Solid:H
xmvc
(z) Broken: H
Pole−zero plot of X (z) / V (z); w / w = 0.07
(s)
d
xmvc
c
0
s
1
−30
0.8
−40
0.6
−50
0.4
−60
−70
100
1000
Fs/2
Phase (radians)
0
Imaginary Part
dB re 1.0 mm/Volt
−20
0.2
0
−0.2
−0.4
−0.6
−pi/2
−0.8
−1
−pi
100
1000
Frequency (Hz)
Fs/2
−1
−0.5
0
0.5
1
Real Part
Figure 2.23: Linear analysis of Eq. (2.85), for adt·1=0.85; left: frequency response, magnitude (upper) and phase
(lower); right: poles and zeros in z-plane.
Chapter 2
68
2.4. Parametric uncertainty
The parameters of a loudspeaker cannot be know a priori from its design and manufacturing
specifications. Parameters of any given loudspeaker will vary from one unit to another from the same
production line due to manufacturing tolerances. Furthermore, the parameters will vary with time due
to ageing, and with temperature variations.
Studies on the parametric variations of loudspeakers due to ambient temperature variations have been
made by Krump (1997) and Hutt (May 2002). Both found the temperature variation to induce
significant changes in the loudspeaker, with the strongest variations found in the suspension stiffness
(kd) and damping (cd). A small variation in the transduction coefficient (φ0) is also found. Significant
variation on the mass md was found by Krump, though not by Hutt. It was suggested by Hutt (June,
2002) that variation in the mass md found by Krump was the result of temperature-induced variations
in the suspension, leading to mass non-uniformity with excursion, as discussed in §2.2.1, part D,
above. It is thus presumed that the variations in mass found by Krump were a ‘large-signal’
(nonlinear) effect, and not due to temperature-variation-induced changes in the small-signal moving
mass. The results of these studies are summarised in Table 2.1, with a key difference for the
suspension stiffness of the microspeaker, discussed below.
Parameter
DC-resistance
Suspension damping
Suspension stiffness
Transduction coefficient
Symbol
Reb
cd
kd
φ0
Temperature Variation
coefficient
Manufacturing tolerance
0.004·Reb·0°C (a)
±10% (b)
-0.05 (c)
±10%(b)
(none)
(d)
-0.005(e)
±30%(b)
n/a
Table 2.1: Known parametric uncertainties due to temperature variation and manufacturing tolerances.
The studies by Krump and Hutt were made on more traditional low-frequency loudspeakers, more
broadly akin to that shown in Figure 2.2 than the microspeaker. With respect to temperaturedependence of the stiffness, the microspeaker has an important difference to the typical loudspeaker.
Specifically, the materials which make up the outer surround and inner spider suspensions are
typically made of soft rubber and woven fabric, respectively. These are materials whose bulk modulii
are known to have strong temperature-dependence. The microspeaker, by contrast nearly universally
uses polycarbonate plastic for its suspension (part (3) of Figure 2.1.) There is no dependence of the
Young’s modulus (and thus suspension stiffness) below 120 °C for polycarbonate plastic, as per Fig.
41 of Nashif and Lewis (1991). Above 120°C, various materials in the microspeaker, such as glues,
insulators, and adhesives, begin to break down, causing irreparable damage. It is, therefore,
considered that this microspeaker will not be used above 120°C. For this reason the temperature
dependence of the suspension stiffness above this temperature is not considered.
(a)
Handbook of Chemistry and Physics, 36th Edition, Chemical Rubber Publishing Co.
(b)
Philips Loudspeaker Systems Telecom Vienna, datasheets 590-N for WD 005XX-series microspeakers.
(c)
Krump (1997)
(d)
For polycarbonate plastic suspension material.
(e)
Krump (1997)
Loudspeaker models
69
2.5. References
Beers, G. L. , and H. Beloar, “Frequency-Modulation Distortion in Loudspeakers,” Proc. IRE 31, 132.
(Apr. 1943)
Birt, David R., “Nonlinearities in Moving-Coil loudspeakers with Overhung voice-coils,” Journal of
the Audio Eng. Soc., 39, pp. 219-231. (April 1991)
Butterweck, H. J., “About Doppler nonlinearities in loudspeakers,” Proceedings of the International
Conference on Acoustics, Speech, and Signal Processing 1989, pp. 2061-2063. (23-26 May, 1989)
Christensen, Knud Bank, and Erling Sandermann Olsen, “Nonlinear Modelling of Low Frequency
Loudspeakers – A More Complete Model,” presented at the 100th Convention of the Audio Eng.
Soc., preprint 4205. (May 11-14, 1996)
Cunningham, W. J., “Non-Linear Distortion in Dynamic Loudspeakers due to Magnetic Effects,” J.
Acoustical Society of America, 21, pp. 202-207. (May, 1949)
Hutt, Steve, “Ambient Temperature Variations on OEM Automotive Loudspeakers”, presented at the
112th Convention of the Audio Eng. Soc., Munich, Germany Convention Paper 5507. (10-13 May,
2002)
Hutt, Steve, personal correspondence. (7 June, 2002)
Jang, Han-Kee, and Kwang-Joon Kim, “Identification of Loudspeaker Nonlinearities Using the
NARMAX Modelling Technique,” Journal of the Audio Eng. Soc. 42, pp. 50-59. (Jan./Feb. 1994)
Kaizer, A. J. M., “Modeling of the Nonlinear Response of an Electrodynamic Loudspeaker by a
Volterra Series Expansion,” Journal of the Audio Eng. Soc., 35, pp. 421-433. (June 1987)
Klapman, S. J., “Interaction Impedance of a System of Circular Pistons”, J. of the Acoust. Soc. Amer.,
11, pp.289-295. (Jan. 1940)
Klippel, Wolfgang, “Nonlinear Large-Signal Behaviour of Electrodynamic Loudspeakers at Low
Frequencies,” Journal of the Audio Eng. Soc. Vol. 40, No. 6 pp. 483-496 (June 1992)
Klippel GmbH, “Large Signal Identification,” Klippel GmbH, Aussiger Str. 3, 01277, Dresden,
Germany. Available online at: http://www.klippel.de/ (Aug. 22, 2001).
Klipsch, Paul W., “Modulation Distortion in Loudspeakers,” J. Audio Eng. Soc. pp. 194-206.
(Presented at the April 29, 1968 34th Convention of the AES) (1968)
Knudsen, M. H., J. Grue Jensen, V. Julskjær, and Per Rubak ”Determination of Loudspeaker Driver
Parameters Using a System Identification Technique,” Journal of the Audio Eng. Soc., 37, pp. 700708. (Sept. 1989)
Krump, Gerhard, “Zur Temperaturabhängigkeit von Lautsprecherparametern,” (“The Temperature
Dependence of Loudspeaker Parameters”) (in German), presented at DAGA-97 (The Acoustical
Society of Germany), (March 3-6, 1997), ISBN 3-9804568-2-X.
Nashif, Ahid D., and Tom M. Lewis, “Data Base of the Dynamic Properties of Materials”, Sound and
Vibration, pp. 14-25. (July 1991)
Newland, D. E., Mechanical vibration analysis and computation, Longman Scientific and Technical,
Burnt Mill, Harlow, England. (1989)
Novak, James F., “Performance of Enclosures for Low-Resonance, High-Compliance Loudspeakers,”
IRE Transactions on Audio, AU-7, pp. 5-13. (Jan.-Feb. 1959)
Olsen, Erling Sandermann and Knud Thorborg, “Diaphragm Area and Mass Nonlinearities of Cone
Loudspeakers,” presented at the 99th Convention of the Audio Eng. Soc., preprint no. 4082. (1995)
70
Chapter 2
Olson, Harry F., Elements of Acoustical Engineering, Second Edition, D. Van Nostrand Co., Inc., New
York, New York. (Sept. 1947)
Olson, Harry F., Acoustical Engineering, D. Van Nostrand Co., Inc., New York, New York. (1957)
Olson, Harry F., “Action of a direct radiator loudspeaker with non-linear cone suspension,” J. Acoust.
Soc. Amer., 16 (1) pp.1-4. (July, 1944)
Oppenheim, Alan V., and Ronald W. Schafer, Discrete-time Signal Processing, Prentice Hall,
Englewood Cliffs, New Jersey, USA. (1989)
Pierce, Allan D., Acoustics: An Introduction to Its Physical Principles and Applications, (1994 Ed.),
Chapter 4, The Acoustical Society of America, Woodbury, New York, U.S.A.. (1994).
Potirakis, S. M. , G. E. Alexakis, M. C. Tsilis, and P. J. Xenitides, “Time-domain nonlinear modelling
of practical electroacoustic transducers.” Journal of the Audio Eng. Soc. 47, pp. 447-468. (June
1999)
Rausch, R., R. Lerch, M. Kaltenbacher, H. Landes, G. Krump, and L. Kreitmeier, “Optimisation of
Electrodynamic Loudspeaker-Design Parameters by Using a Numerical Calculation Scheme”,
ACUSTICA-acta acustica, 85, pp. 412-419. (1999)
Small, Richard H., “Direct-Radiator Loudspeaker System Analysis,” IEEE Transactions on Audio and
Electroacoustics, Vol. AU-19, pp. 269-281 (Dec. 1971)
Small, Richard H., “Closed-Box Loudspeaker Systems, Part I: Analysis,” Journal of the Audio Eng.
Soc., 20, pp. 798-808. (Dec. 1972)
Small, Richard H., “Closed-Box Loudspeaker Systems, Part II: Synthesis,” Journal of the Audio Eng.
Soc., 21, pp. 11-18. (Jan-Feb. 1973)
Small, Richard H., “Vented-Box Loudspeaker Systems – Part 1: Small-Signal Analysis,” Journal of
the Audio Eng. Soc., 21, pp. 363-372. (Jun. 1973)
Small, Richard H., “Vented-Box Loudspeaker Systems – Part 2: Large-Signal Analysis,” Journal of
the Audio Eng. Soc., 21, pp. 483-444. (Jul. 1973)
Small, Richard H., “Vented-Box Loudspeaker Systems – Part 3: Synthesis,” Journal of the Audio Eng.
Soc., 21, pp. 549-554. (Sep. 1973)
Small, Richard H., “Vented-Box Loudspeaker Systems – Part 4: Appendices,” Journal of the Audio
Eng. Soc. 21, pp. 635-639. (Oct. 1973)
Small, Richard H., “Loudspeaker Large-Signal Limitations,” presented at the 1st Australian Regional
Convention of the Audio Eng. Soc., preprint no. 2102. (Sept 25-27, 1984)
Thiele, A. N, “Loudspeakers in Vented Boxes,” Proceedings of the IRE Australia, 22, pp. 487-508
(Aug. 1961); reprinted in Journal of the Audio Eng. Soc., 19, pp. 382-392. (May 1971)
Vanderkooy, John, “A Model of Loudspeaker Driver Impedance Incorporating Eddy Currents in the
Pole Structure,” Journal of the Audio Eng. Soc., 37, pp. 119-128. (Mar. 1989)
Zóltogórski, Bonislaw, “Moving Boundary Condition and Non-Linear Propagation as the Sources of
Non-Linear Distortions in Loudspeakers,” presented at the 94th Convention of the Audio
Engineering Society, preprint no. 3510. (16-19 Mar, 1993)
71
3.
Theory of active control of loudspeakers
This chapter presents theory of systems for active control of loudspeakers. Active control is
considered in the context of three possible architectures, feedback, feedforward, and adaptive
feedforward, as they were presented in the introduction.
Several different feedback systems are discussed. Current feedback and negative output impedance
amplifiers are first discussed, and a historical view of their development is given. ‘Motional feedback’
systems for active control, wherein a measured vibration signal is used for feedback, are then
reviewed. Implementation problems of these feedback systems are discussed.
Feedforward processors are presented at a general level. Both linear and nonlinear feedforward
processing is considered.
A special section on the theory of feedback linearisation is presented, along with a discussion on how
this can be used to develop a nonlinear feedforward controller for a loudspeaker.
The theory of feedback linearisation is applied to the nonlinear discrete time model developed in
§2.3.4, above. This leads to a simple algorithm for compensation of loudspeaker nonlinearities.
As discussed in §2.4, certain properties of the loudspeaker cannot be known a priori, as they are
subject to drift due to various factors. It is for this reason necessary to make any feedforward
controller adaptive, so as to be properly tuned to the loudspeaker. This leads to the adaptive
feedforward controller, mentioned in the introduction, and shown in Figure 1.4. The adaptive part of
this type of controller is performed using adaptive signal processing. A brief review of adaptive signal
processing theory, and a special discussion on adaptive recursive (IIR) filters, is therefore presented at
the end of this chapter.
72
Chapter 3
3.1. Feedback control for loudspeakers
The improvement of loudspeakers by electrical means has been actively researched, more or less in
parallel with research on the loudspeaker itself. The historical and technical context of feed-forward
equalisation and nonlinear compensation lies in theory and design of feedback controllers. To this
end, such well-known systems as negative output impedance amplifiers and motional feedback are
reviewed.
3.1.1. Constant-current output amplifiers
A constant-current output amplifier offers a few advantages over the typical constant-voltage (lowimpedance) power amplifiers. This can be seen, from a theoretical point of view, in a straightforward
manner by inspection of the nonlinear differential equation pair of (2.50) and (2.51). If the coilcurrent ic(t) is held constant by the power amplifier, the nonlinear terms in the voltage equation (2.50)
will not cause nonlinearity between the input and output (as they would for a constant-voltage
amplifier). Only nonlinear terms in the force equation, (2.51) will generate nonlinear distortion.
Perhaps more significantly, any changes in the DC resistance Reb due to heating of the voice-coil
(known as ‘power compression’) will not effect the resulting response. These benefits were used as
arguments in favour of using constant-current output amplifiers by Mills and Hawksford (1989).
The primary complication in using a current-drive amplifier is that the electrical damping is lost,
resulting in an excess Qtc. Mills and Hawksford solved this problem by using some velocity feedback,
wherein the velocity signal was obtained by a secondary winding on the voice-coil. Transformer-like
coupling between the main drive coil and this sensing coil was compensated for by an appropriate
network. The constant-current output amplifier does not seem to have received any interest since the
work of Mills and Hawksford.
3.1.2. Negative amplifier output impedance
Before the negative feedback amplifier was formalised as a method for reduction of amplifier
distortion, various feedback systems were proposed using some kind of interface to the loudspeaker.
The first record of these is from Voigt (1925), whose patent describes the modified Wheatstone bridge
shown in Figure 3.1. The bridge is connected between the amplifier and loudspeaker, producing a
signal proportional to the diaphragm velocity. The velocity-analogous signal may be fed back into the
input to the amplifier, thereby damping the loudspeaker’s resonance. This was seen as advantageous,
as mechanical damping of this resonance would result in an efficiency loss, and additional passive
electrical damping would require a larger magnet resulting in a higher cost and weight.
Theory of active control of loudspeakers
73
Bridge
Amplifier
output voltage
k R1
R1
Loudspeaker
Reb
Leb
vu(t)= 0ud(t)
k Reb
To amplifier
input
ud(t)
ic(t)
vamp(t)
vc(t)
vu(t)
fc(t)
Zm(s)
0ic(t)=fc(t)
a vu(t)
Figure 3.1: Basic diagram of the impedance bridge for generating velocity feedback
signal to create a negative output impedance amplifier.
This principle has been written about by many authors over many years. It was described by Yorke
and McLachlan (1951) who called it an ‘Amplifier of Variable Output Impedance’, Clements (1951,
1952) who called it ‘Positive Feedback’, Wentworth (1951) who called it ‘Inverse Feedback’, Childs
(1952) who called it ‘Dynamic Negative Feedback’, and Wilkins (1956) who referred to it as ‘Control
of Amplifier Source Resistance’. Agreement seemed to have come upon the term ‘Negative Output
Impedance’, used by Werner (1957), Werner and Carrell (1958), Steiger (1960), Thiele (1961)1, Ståhl
(1981), and Normandin (1984). However, this system was simply referred to as the ‘bridge-version’
of motional feedback in papers by de Boer (1961), Klaassen and de Koning (1968), and Adams and
Yorke (1976). Furthermore, on the commercial market it was referred to as ‘damping control’, a
popular feature in the 1960’s, seen now only in the idiosyncratic electric-guitar amplifier market.
Perennial terminology confusion was ensured in papers by Holdaway (1963), who referred to this as
‘Velocity Feedback,’ and Birt (1981), who demonstrated true creativity where one might have thought
it exhausted with the name ‘Load-Adaptive Source Impedance.’ Little new was contributed to the
subject by these latter authors, with the notable exception of Ståhl (1981) and Normandin (1984) who
used it to interesting effect, which will be discussed below.
It is also possible to derive a velocity signal using the same principle as described above, but instead
using active analogue electronics. Such a system was described by Bai and Wu (1999). This was used
to generate a feedback signal for a digital feed-forward controller to provide linear adaptive
equalisation. The work of Bai and Wu was developed in the context of control theory, wherein the
analogue circuitry providing the velocity signal is referred to as an observer model.
The impedance bridge shown in Figure 3.1 can only provide a velocity-analogous signal to the
controller under the following two conditions:
• The resistor in the lower-left leg maintains the value kReb
• The transduction coefficient φ0 remains uniform with respect to diaphragm-coil displacement.
These two conditions are not always met. The actual DC resistance will change with coil temperature,
which is affected by ambient temperature, and with heating due to thermal dissipation. At large
1
This is mentioned in an off-hand manner in section 12 of Thiele’s famous 1961 paper (better known by its 1971
re-publication in the J. of the Audio Eng Soc.) the main subject of which was the application of analogue
active-filter alignment tables to the design of vented (ported) enclosures.
Chapter 3
74
displacements the transduction coefficient will not be uniform with respect to displacement, as
discussed in §2.2.1, part A.
A further problem is that the velocity-analogous signal is proportional to the system output, the
acoustic pressure, only if the loudspeaker is mounted in a closed box, wherein the acoustic pressure is
given by the time-derivative of the diaphragm velocity as per (2.24). If the loudspeaker is mounted in
a ported or horn-loaded enclosure, the relationship between velocity and pressure will be more
complex. For this reason, the negative-output impedance amplifier is not a closed-loop control system
in the strict sense of that shown in Figure 1.2.
Whether the system of Figure 3.1 may be described as ‘motional feedback’ also seems to be a matter
of disagreement in literature. Adams (1979) firmly asserts “[this system] can rightly be considered as
… the use of motional feedback.”1 The opposite is equally firmly asserted by Ståhl (1981) in his
conclusions about a special application of a negative output impedance amplifier, wherein he states:
[The present system] is not a feedback system. In [a motional feedback] system, the output signal
from the loudspeaker is sensed and in one way or another fed back to the amplifier. [The present
system] instead uses an amplifier with a special output impedance to which the loudspeaker is
connected.2
Confusing nomenclature aside, perhaps the most interesting use of the negative output impedance
resulting from Figure 3.1 was the ‘Amplifier Controlled Euphonic Bass’ (ACE) amplifier described by
Ståhl (1981). In addition to using this circuit to ‘cancel’ the effects of the DC-resistance, Ståhl
described an amplifier wherein passive components are connected in parallel to the output of the
amplifier’s terminals. This system removed the effect of the loudspeaker’s components, allowing the
response to be controlled by selection of values of the passive components connected in parallel with
the amplifier output. This technique found some commercial success in subwoofers which continues
to this day.
Both the negative output-impedance amplifier and the ACE amplifier described above suffer from a
tuning problem. Proper operation of both systems requires that the resistor kReb in the lower left leg of
the bridge in Figure 3.1 be tuned to Reb, the blocked electrical resistance of the loudspeaker. This is
problematic, because the value of Reb will change with coil temperature, which will change with
ambient temperature and with heating due to resistive electrical power dissipation. The change in Reb
due to this temperature variation will be on the order of –30 / + 60%. The effect of such changes in
Reb on the frequency response of an ACE amplifier-powered loudspeaker was found by Lechevalier
(2000) to be between ±2dB to ±10dB, dependent upon frequency.3
3.1.3. Feedback processing using vibration measurement
The first publication describing what is more commonly thought of as motional feedback appeared in
Hanna (1927), wherein a secondary coil on a traditional loudspeaker served as an electrodynamic
velocity sensor, the output of which was fed back into the input of an amplifier. An example of an
electrodynamic loudspeaker with a suitable additional sensing coil is shown in Figure 3.2. The same
1
p. 68, paragraph 5 of Adams (1979).
2
p. 595, paragraph 3 of Ståhl (1981); it should be noted that Ståhl fails to point out that the negative impedance
amplifier used in his system is equivalent to the ‘bridge-type MFB’ system Ståhl discusses subsequently on
same page of his 1981 paper, in paragraph 5.
3
Figure 68, p. 60 of Lechevalier (2000).
Theory of active control of loudspeakers
75
system was described by Olson (1947),1 Tanner (1951), and reviewed in a general discussion on
motional feedback by Klaassen and de Koning (1968).
Secondary
Magnetic Circuit
Secondary Magnet
Sensing Coil
Extended Voice
Coil Former
Pole-piece mount
Primary (drive)
voice coil
Figure 3.2: Example of an electrodynamic loudspeaker with a secondary magnet system and coil serving as an
electrodynamic velocity sensor.
The obvious disadvantage of the system show in Figure 3.2 is that it requires a secondary magnetic
system, which will increase the cost and weight of the loudspeaker. A less expensive strategy is to add
a secondary winding to the primary coil. This principle was first discussed in an application for
microphones by de Boer and Schenkel (1948). This method has the problem that there will be
transformer-coupling between the drive and sensor coils, as noted by Tanner (1951). This problem
can be compensated for by additional coils, or electronically by an analogous model using active
analogue electronics as was done by Mills and Hawksford (1989).
The first motional feedback system using the signal from an inertial accelerometer was described by
Klaassen and de Koning (1968). These authors describe a system using an accelerometer mounted
upon the dust cap (part 2b in Figure 2.2.) A distinct advantage of acceleration feedback is that, if the
loudspeaker is mounted in a closed box, the shape of the frequency response of the acoustic pressure
will be the same as the acceleration response. Thus the acceleration signal permits a more direct
implementation of the closed-loop feedback system of Figure 1.2 than the velocity feedback systems.
Unfortunately, due to the high cost of suitable accelerometers, commercial success of this technique
has been limited, and now appears only in domestic ‘sub-woofers’ from a one manufacturer (Hall,
1989).
Other combinations of current, velocity, and acceleration feedback have been described in various
papers. Greiner and Simms (1984) describe one such system, wherein a current and accelerometer are
used to provide a fairly flat frequency response system, with moderate distortion reduction over the
same unit driven by a constant-voltage output amplifier. Catrysse (1985) described a feedback system
using a combination of current and velocity feedback, though in this case the velocity signal was
obtained by differentiating a displacement signal obtained by a capacitive sensor. These systems have
not received further interest since the publication of these papers, presumably due to the excess cost of
the feedback sensors.
1
pp. 158-159 of Olson (1947); also appears in the more widely available Acoustical Engineering, Olson (1957),
pp. 168-169.
Chapter 3
76
3.2. Feedforward controllers
3.2.1. Linear feedforward processing
Linear feedforward processing for loudspeakers has been in use for many years. Equalisation, by
digital, and by passive and active analogue filters has been written on at length in literature, and
widely successfully commercialised. Perhaps the most widespread application of linear feedforward
equalisation of loudspeakers is the use of sixth-order vented box alignments described by Thiele
(1961), among other authors.
Analogue passive and/or active components can be used to extend the low-frequency range of a
loudspeaker, forming one of the simplest ‘feedforward processors’ for a loudspeaker. An informative
review of these systems is given by von Recklinghausen (1985). Active analogue filters for equalising
the low-frequency response of a loudspeaker has been shown as a simple method to control the
overall-Q value and cut-off frequency of a loudspeaker mounted in a closed box (Leach, 1990). (This
type of linear feedforward processing is studied further in Chapter 5.)
Frequency equalisation of the far-field response of a loudspeaker is a more sophisticated application of
linear feedforward processing. The tactic has generally been to process the audio signal before sent to
the loudspeaker with some filter having a response approximating the inverse of a measured response
of the loudspeaker. Some research has focused on only low-frequency equalisation (Greenfield and
Hawksford, 1991). Other research has been made on filter design for full-frequency equalisation
(Karjalainen et al., 1999.) More research still has considered equalisation of the complete
loudspeaker-room system (Kirkeby and Nelson, 1999.) Unfortunately, these latter projects and others
similar to them apparently did not fully comprehend a statement in Greenfield and Hawksford’s 1991
paper, explaining:
The on axis measurement appears to be an obvious choice [for the response to be inverted]. But
bearing in mind the requirement for [equalisation] over a listening space, the value of correcting
small aberrations that occur in the on-axis response and do not occur [in] the off-axis response
seems suspect. Indeed, this form of [equalisation] may prove detrimental to off-axis responses.
As mentioned in the introduction, the sound field produced by the loudspeaker will vary with different
positions from the loudspeaker. For this and other reasons, detailed correction of the far-field
response of a loudspeaker is not considered in this thesis.
3.2.2. Nonlinear feedforward processing
Perhaps the first discussion on feedforward compensation of loudspeaker nonlinearity appears in a
paper by MacDonald (1959). This paper discusses how to compensate for simple distortion
mechanisms such as squarers or cubers via pre- or post-distortion. Although the suggested application
is for a loudspeaker, the paper does not consider specifics of applying this method to the loudspeaker,
wherein the squarers and cubers would need to be made frequency dependent. The next step in such
feedforward distortion compensation was not taken until over three decades later by Birt (1991),
wherein nonuniformity of the transduction coefficient was compensated via a look-up table. The
diaphragm displacement was measured by a capacitive sensor of the same type used by Catrysse
(1985), described above. These methods for feedforward distortion compensation did not spur much
commercial nor academic interest. This is presumably due to the complexity of the hardware, or
processing, or both.
More recently, feedforward nonlinear processors for loudspeakers have been developed using the
theory of feedback linearisation. Although, by its name, feedback linearisation was conceived as a
method for designing feedback controllers, several successful adaptations have led to pure
feedforward controllers. The theory of feedback linearisation and its application to loudspeakers is
discussed in detail in the next section.
Theory of active control of loudspeakers
77
3.3. Feedback linearisation
Feedback linearisation, also called the Differential Geometric Method for Nonlinear Control, is a
general, abstract, complete formal theory for the control of nonlinear dynamic systems. Although the
word ‘feedback’ forms part of its name, it is not a pure closed-loop control system in the classical
sense of a feedback servo controller. The method uses elements of feedforward control, in that it
utilises a complete model of the system’s dynamics, and feedback control, as it uses a measurement
(‘feedback’) of the system’s state. Together, these provide linear and nonlinear control of the plant’s
dynamics.
Feedback linearisation was developed in the late 1980’s in the field of Automatic Control.
Presentations of this theory have been published in textbooks by Isidori (1989), Nijmeijer and Schaft
(1990), Slotine (1991), and Vidyasagar (1993). A concise review of the theory was presented in the
introduction to a Ph.D. thesis by Vesterholm (1995). The presentation by Nijmeijer and Schaft is by
far the most rigorous and pedantic, and thus unlikely to be understood by those without a thorough
background in both control theory and differential geometry.
It is not clear who first applied feedback linearisation to loudspeakers. Beerling et al. (1994) cite a
reference for application of feedback linearisation to a loudspeaker dated April 1992 (Suykens et
al.,1992), wherein an inverse dynamics processor uses feedback signals from a model of the
loudspeaker (observer model) to create a feed-forward distortion compensation processor. However,
this reference gives no bibliographical information other than ‘Leuven.’ It is inferred by your author
that this is an internal publication of Katholiek Universiteit Leuven, in Belgium, though this is not
certain. Five months later in September of that year, Wolfgang Klippel published a now well-known
paper describing a ‘Mirror Filter’ (Klippel, 1992). Klippel’s mirror filter operates in the same manner
as the inverse-dynamic processor, with the exception that the loudspeaker’s behaviour is not modelled
with a distinct observer model. Instead, the loudspeaker’s dynamics are simulated by direct processing
of the audio input signal, leading to a controller which is overall more simple than that using an
observer model (as in Berling et al.). Appropriately, the terms ‘inverse dynamics processor’ and
‘mirror filter’ can generally be understood to describe the same thing. Klippel did not, however, make
reference to feedback linearisation theory, though this is not surprising, as at that time it was quite
unknown in the audio and acoustics community wherein most studies on loudspeakers were made. All
of this led to the similarity between the mirror filter and inverse dynamics processor to be missed, the
clearest example of which appears in a paper by Schurer et al. (1998), a comparison study that
considered the mirror filter to be in an entirely different category from methods based on feedback
linearisation. It was finally recognised by Klippel (Nov., 1998) that the mirror filter is simply a feedforward case of the integrator-decoupled form of feedback linearisation, wherein the system states are
determined by processing of the input to the compensator, instead of by an observer model processing
the output of the compensator. This difference is discussed in detail in §§3.3.4 and 3.3.5.
Feedback linearisation is based on a state-space description of the dynamics of the plant-to-becontrolled, which in this case is the loudspeaker. State-space descriptions are significantly different
from the classical descriptions of loudspeaker dynamics. Thus the state-space representation and its
role in DSP-based distortion compensation are first reviewed.
3.3.1. Feedback linearisation of continuous-time systems
Consider a general class of systems that may be described by this first-order differential equation:
x& (t ) = f (x(t ) ) + g(x(t ) ) u (t )
y (t ) = h(x(t ) )
(3.1)
Chapter 3
78
The system feedback function f, the input function g, and the output function h are all considered to be
nonlinear functions of the state vector x(t). A block diagram of such a system is shown in Figure 3.3.
u(t)
.
g(x)
Σ
f(x)
x(t)
h(x)
∫
y(t)
x
Figure 3.3: State-space representation of a nonlinear system.
The basic goal of linearisation is to create a linear relationship between some input signal and the
output y(t) in (3.1). In the formal theory of Feedback Linearisation, this is done by applying a
coordinate transformation to the state vector x(t). The idea is to find a different state vector such that
the operators f, g, and h are not nonlinear. The formal theory of coordinate transformation of the state
vector is not likely to be familiar to the audio, signal processing, or loudspeaker engineer. As it is not
necessary for comprehension of how feedback linearisation is used for loudspeaker distortion
compensation, it is thus not explained nor used in this thesis.
The essential feature of feedback linearisation is that by taking sufficient numbers of the total time
derivative of the output y(t), given certain conditions, one will eventually arrive at an expression that
depends explicitly on the input. Such an expression can be inverted by simple algebraic manipulation.
This inverted expression may then be used as a ‘control law,’ i.e. the basis for a controller
(compensator), which will cancel-out (compensate) the nonlinearity in the system. As per §2.3 of
Vesterholm (1995), the total time derivative of the output y(t) of the system described by (3.1) may be
represented as
∂h(x(t ) )
x& (t )
∂x(t )
∂h(x(t ) )
(f (x(t ) ) + g(x(t ) ) u (t ) )
=
∂x(t )
y& (t ) =
(3.2)
According to the theory of differential geometry, (3.2) may be re-written as the Lie derivative of
h(x(t)) along the vector fields f(x(t)) and g(x(t)), using the notation
y& (t ) = L f h(x(t ) ) + L g h(x(t ) ) u (t )
(3.3)
The Lie derivatives are calculated from the dot-product (inner-, or scalar-product) between the vectors
∂h(x(t ) ) dx(t ) and f(x(t)) and g(x(t)), i.e.
∂h(x(t ) )
• f (x(t ) )
∂x(t )
∂h(x(t ) )
L g h(x(t ) ) =
• g(x(t ) )
∂x(t )
L f h(x(t ) ) =
(3.4)
If L g h(x(t ) ) ≠ 0 , then the form of (3.3) will provide an explicit formula for the output y(t) in terms of
the input u(t). If, instead, L g h(x(t ) ) = 0 , then it will be necessary to take a higher-order time
derivative of the output in order to obtain an explicit formula. Assuming this is so, the second
derivative of the output is taken and represented as:
&y&(t ) = L(f2) h(x(t ) ) + L g L f h(x(t ) ) u (t )
(3.5)
Theory of active control of loudspeakers
79
If the term L g L f h(x(t ) ) is zero, then it will again be necessary to obtain a still higher-order time
derivative. Given certain conditions, one will eventually find a derivative of the output for which the
term in which the input u(t) appears will not be zero. By defining the order of the derivative at which
this occurs as r, it may be said that the system concerned has a relative degree of r. The rth-derivative
of the output may be expressed as
y ( r ) (t ) = L(fr ) (x(t ) ) + L g L(fr −1) h(x(t ) ) u (t )
(3.6)
As (3.6) does depend directly in the input u(t), it can be used to form the basis of a nonlinear
controller, or compensator. This controller will process the signal v(t) according to the inverse of
(3.6), and feed this to the input to the plant. Thus the output of the controller (fed to the input to the
plant) is processed as so:
u (t ) =
v(t ) − L(fr )h(x(t ) )
Lg L(fr −1)h(x(t ) )
(3.7)
the effect will be to create a linear relationship between the input v(t) and the output y(t) as so
y ( r ) (t ) = v(t )
L{}
y ( s) 1
=
v( s ) s r
⇒
(3.8)
where r is the relative degree of the system, defined above.
‘Inverse Dynamics’ Nonlinear Controller
v (t )
+
Σ
u(t)
×
-
Plant (nonlinear)
.
g(x)
L (fr ) h (x )
L g L(fr −1) h(x )
Σ
f(x)
x(t)
h(x)
∫
y(t)
x
−1
state feedback
Figure 3.4: Block diagram of feedback linearisation control system.
The relationship in (3.8), between the system input and the output created by the inverse-dynamic
controller, may not be desirable. The small-signal behaviour of the system, i.e. that which can be
described by a linear model, may, in fact, be the desired input-output relationship. (This is particularly
the case for the loudspeaker, as will be discussed below.) When this is the case, it is necessary to ‘reintroduce’ the linear dynamics, by pre-filtering the signal v(t).
It is assumed that for small-signals, nonlinearity in the plant may be neglected, and its input-output
may be described by a linear transfer function as so:
H ld ( s ) =
y(s)
u (s)
(3.9)
Chapter 3
80
The linear dynamics may be ‘re-introduced’ by pre-filtering the input to the inverse-dynamics
controller v(t) by a linear filter with the frequency response of the linear dynamics of the system, i.e.
Hld(s) in (3.9). This is shown in block form in Figure 3.5. Note that with the linear dynamics ‘reintroduced’ in this way, the input to the controller is defined as w(t), instead of v(t).
Nonlinear compensating processor
w (t )
Linear dynamics
‘re-introduction’
H ld (s)
Plant (nonlinear)
‘Inverse Dynamics’ Processor
v( t )
+
Σ
u(t)
×
-
L (fr ) h (x )
L g L(fr −1) h(x )
.
g(x)
Σ
f(x)
x(t)
h(x)
∫
y(t)
x
−1
state feedback
Figure 3.5: Feedback linearisation control system, with re-introduction of linear dynamics, providing
compensation only of nonlinear dynamics.
3.3.2. Example: simplified closed box loudspeaker in continuous-time
We consider the state-space representation of the loudspeaker’s dynamics where the input u(t) is
defined as the voltage-drop across the loudspeaker’s terminals:
x& (t ) = f (x(t ) ) + g(x(t ) ) u (t )
y (t ) = h(x(t ) )
(3.10)
As discussed above, the formal theory of feedback linearisation was first applied to a loudspeaker by
Suykens et al. (1995,) in the paper mentioned in the introduction to this chapter. In Suykens’ 1995
paper, the details of applying the theory were worked out for a loudspeaker excluding any effects of
acoustic loading. This was extended to the case of a loudspeaker in a vented cabinet (a single
acoustic resonator) by Schurer (1997.) In the present example, we consider the case of a loudspeaker
with simple acoustic loading, and with negligible electrical inductance. With these considerations, the
system, input, and output vector fields are as follows:
x2 (t )





f (x(t ) ) = 
 kt (x1 (t ) )

1 
φ2 (x1 (t ) ) 
 x2 (t ) 
x1 −  ct +
−

mt
mt 
Reb 


 0 



g (x(t ) ) = 
 φ(x1 (t ) ) 
 mt Reb 
 x (t )
h(x(t ) ) =  1 
 0 
where the state vector x(t)is defined on ℜ2 as so
(3.11)
Theory of active control of loudspeakers
 x (t )   x (t )   x (t )
x(t ) =  1  =  d  =  d 
 x2 (t ) ud (t )  x&d (t )
81
(3.12)
Other terms in (3.11) and (3.12) are as defined previously in §§2.1-2.2. Note that output y(t) is
defined as the diaphragm displacement xd(t). The diaphragm displacement is a suitable output, from
the perspective of developing a feedback linearisation algorithm. However, from the perspective of
evaluating the electro-acoustic performance of the loudspeaker, it is more interesting to analyse the
acoustic pressure. The relationship between displacement and acoustic pressure is well-studied; see
Small (1971).
The control law for a nonlinear compensator for the loudspeaker described by (3.10)-(3.12) is
developed using the method described above to obtain an expression for the output y(t) which depends
explicitly on the input, which in this case is u(t). The first time derivative of the output is given by
y& (t ) = L f h(x(t ) ) + L g h(x(t ) ) u (t )
= [1 0] • f (x(t ) ) + [1 0] • g(x(t ) ) u (t )
= x 2 (t )
(3.13)
The input u(t) does not appear in the RHS of this expression. Thus it is necessary to take the next
higher-order derivative of the output y(t) with respect to time:
∂y& (t )
∂y& (t )
f (x(t ) ) +
g(x(t ) ) u (t )
∂x(t )
∂x(t )
= [0 1]f (x(t ) ) + [0 1]g(x(t ) ) u (t )
&y&(t ) =
=−
(3.14)
k t (x1 (t ) )
φ( x1 (t ) )
φ (x1 (t ) ) 
1 
 x 2 (t ) + u (t )
x1 (t ) −
c
+
t


mt Reb
Reb 
mt
mt 
2
As this expression does depend directly on the input u(t) it can be used to derive a control law for
linearisation (compensation of nonlinearity.) The control law for linearisation is simply the inverse of
the relationship between u(t) and &y&(t ) in (3.14). From the practical standpoint of implementing a
controller with some input signal, we consider the input to this controller as v(t). The controller
operates on the input v(t) according to:
u (t ) =
Reb
φ( x1 (t ) )



φ 2 (x1 (t ) ) 
 x 2 (t ) + mt v(t ) 
k t (x1 (t ) ) x1 (t ) +  c t +
Reb 



(3.15)
The ‘control law’ for the controller, given in (3.15), creates the following relationship between v(t) and
xd(t):
&x&d (t ) = v(t )
L
⇒
x d ( s) 1
= 2
v( s )
s
(3.16)
Notice that (3.16) is equivalent to saying that the diaphragm acceleration will have a unity transfer
function with respect to the input signal v(t). Although this does have the advantage of eliminating
nonlinear behaviour of the loudspeaker, it is not the desired linear input-output frequency response.
At low frequencies, this response would generate very large displacement, thereby generating very
large nonlinearities - beyond that which could be realistically controlled.
For a loudspeaker mounted in a closed-box, the desired linear acceleration frequency response is, as
discussed in §2.1.7, a second-order high-pass filter. This desired response may be restored in the
controller by pre-filtering the input to the nonlinear control law v(t) by an appropriate low-pass filter.
By defining the input to this filter as w(t), such a filter should have the following transfer function:
Chapter 3
82
H ld ( s ) =
v( s )
w( s )
1
= 2 2
s ω0 + s ω0 Qtc + 1
(3.17)
This filter ‘pre-emphasises’ v(t), the input to the inverse dynamics processor. Given that the inverse
dynamics processor eliminates all behaviour of the plant, i.e. both linear and nonlinear dynamics, the
pre-emphasis the filter of (3.17) represents a re-introduction of the linear dynamics of the plant.
3.3.3. State observer and partial state measurement
The controller developed using the theory of feedback linearisation described in §3.3.1 above suffers
from the same problem as traditional feedback processors described in §3.1 above. Namely, the
controller developed according to feedback linearisation requires measurement of the state vector x(t).
In the example of feedback linearisation applied to the loudspeaker mounted in a closed box described
above, the state vector comprises the diaphragm displacement xd(t) and velocity ud(t). As per the
discussion on traditional feedback systems, directly measuring the displacement and velocity is either
impractical or expensive, or both.
One solution to this problem is to make a partial state measurement, i.e. to measure one state, and
simulate the other states using a state observer. One example of how this can be done was presented
by Beerling et al. (1994). In the system presented by Beerling et al., an inertial accelerometer is
placed on the loudspeaker diaphragm. The state observer integrates and double integrates this signal
from the accelerometer, and also calculates the voice-coil current ic(t) (needed in that case for
compensation of nonlinearity caused by Leb nonuniformity). The distinct disadvantage of this
approach, for the microspeaker, is that no suitably inexpensive and lightweight accelerometer is
available.
Another possibility would be to measure the voice-coil current, as per the arrangement shown in
Figure 3.6. In this case, a state observer would simulate the state vector with an appropriate model of
the loudspeaker. As it happens, there is problem rendering impractical any such approach wherein a
state observer predicts the state vector from a measured signal.
There is a time-delay inherent to all sigma-delta A/D and D/A converters. Sigma-delta converters are
used in all modern digital audio systems, due to their low cost relative to their frequency-resolution
product. These converters use linear-phase interpolation and decimation filters for converting the
high-frequency one-bit stream to the audio bandwidth full resolution (typically 16 bit) data stream.
Such filters create a delay of some 20 or so samples (Zölzer, 1994), the exact number depending on
the design of the interpolation filter. Thus the delay in the complete ‘round-trip’ caused by D/A and
A/D conversion would be some 40 samples. This would make the state predicted by the state observer
of Figure 3.6 some 40 samples ‘late’ for the input signal v(t). For audio systems, this effect limits the
bandwidth of this type of controller such that it is impractical.
Theory of active control of loudspeakers
83
Nonlinear compensating processor
w( t )
Linear dynamics
‘re-introduction’
H ld (s)
Plant (nonlinear)
‘Inverse Dynamics’ Processor
v(t)
+
Σ
u(t)
×
-
L (fr ) h (x )
.
g(x)
Σ
x(t)
y(t)
x
f(x)
L g L(fr −1) h(x )
h(x)
∫
−1
ic(t)
partial state
measurement
state estimate
State observer
g o(xˆ )
Σ
f o (xˆ )
x̂&
∫
x̂
Figure 3.6: State vector estimation based on partial state measurement, e.g. ic(t), and a state observer.
3.3.4. Feedforward formulation using a ‘simulation’ state observer
A feedback linearisation-based controller presented by Schurer et al. (1997) employed a state observer
which made no measurement on the plant whatsoever. In this system described by Schurer et al., the
input to the state observer was the output from the controller u(t), as per Figure 3.7. This is
theoretically possible, as the to the system vc(t) – and with an appropriate and accurate model of the
system, the state vector can be predicted from the output of the controller u(t). In this case, the
feedback linearisation controller uses no signal from the actual plant. This results in a controller of the
pure (non-adaptive) feedforward type, as per Figure 1.3.
Chapter 3
84
Nonlinear compensating processor
Linear dynamics
‘re-introduction’
w (t )
H ld (s)
Plant (nonlinear)
‘Inverse Dynamics’ Processor
v( t )
+
Σ
u(t)
×
-
.
g(x)
L (fr ) h (x )
Σ
f(x)
L g L(fr −1) h(x )
x(t)
h(x)
∫
y(t)
x
−1
state estimate
State observer
gˆ ( xˆ )
Σ
fˆ (xˆ )
x̂&
hˆ ( xˆ )
∫
x̂
Figure 3.7: State observer used to determine x̂ , an estimate of the system’s state vector.
Schurer implemented the complete controller with digital processing. The digital processor, operating
in discrete time, cannot directly implement a model of the loudspeaker which is expressed in
continuous time with differential operations.1 Schurer’s solution to this problem was to form his
model of the loudspeaker in the state-space first-order differential equation of (3.2), and solve this
equation by numerical integration. In this way, the new value of the state vector is predicted from the
existing value of the state vector and its derivative as so
x[n] =
N
∑
j =1
α j x[n − j ] +
M
∑β
j =1
& [n
jx
− j]
(3.18)
where αj and βj are coefficients defined by numerical integration rules .
One disadvantage to this approach is that, due to the feedback loop created between u(t), the estimated
state xˆ [n] , and the use of this estimated state in the inverse dynamics processor, it can become
unstable.
3.3.5. Feedforward formulation assuming ideal alignment
According to the theory of feedback linearisation, if the parameters of the inverse dynamics processor
of Figure 3.4 are tuned to the actual plant with reasonable accuracy, the state vector can be simulated
from the input to the nonlinear controller v(t) in a more simple manner than using the state observer
described in §3.3.4. This is a result of (3.8), stating that, if the Inverse Dynamics processor is
accurately tuned to the plant, the rth derivative of the output will be given by the input to the nonlinear
controller, v(t).
1
This problem was discussed at a general level in §2.3, and is the motivation for developing the discrete-time
models of the loudspeaker developed in that section.
Theory of active control of loudspeakers
85
The simplification provided by this method is easily demonstrated for the case of the closed-box
loudspeaker, ignoring Leb, described in §3.3.2. According to (3.16), the first system state x1(t) = x(t)
will have its second time derivative directly equal to the modified control input v(t). Therefore,
assuming the feedback linearisation law in (3.15) operates properly, the system states can be
determined by integration of the input to the Inverse Dynamics controller, v(t).
As mentioned above, integration cannot be performed directly in a digital implementation. It is
necessary to perform a numerical approximation to the integration. The system states are, therefore,
approximated as so:
x1 [n] =
N
∑α
j x1 [ n
− j] +
j =1
x 2 [ n] =
M
∑β
− j]
(3.19)
∑ β v[n − j ] .
(3.20)
j x 2 [n
j =1
N
∑α
j x 2 [n
− j] +
j =1
M
j
j =1
A block diagram of the controller wherein the states are computed as above is shown in Figure 3.8.
Nonlinear compensating processor
w (t )
Linear dynamics
‘re-introduction’
H ld (s)
Plant (nonlinear)
‘Inverse Dynamics’ Processor
v( t )
+
Σ
u(t)
×
-
L (fr ) h (x )
L g L(fr −1) h(x )
.
g(x)
Σ
f(x)
x(t)
h(x)
∫
y(t)
x
−1
∫
∫
..
x̂
∫
Figure 3.8: State calculation by direct integration of input to nonlinear controller.
Note that if Leb is to be considered in the loudspeaker, the state vector will include the voice-coilcurrent ic(t). The voice-coil current cannot be determined from the input to the nonlinear controller
v(t) by simple integration. Instead, it is necessary to predict the current using the linear voltage
equation of (2.1), as explained by Klippel (1992). Given that the state vector will invariably include
the diaphragm velocity ud(t), that the voice-coil voltage will also generally be known, and that there
are no memory-dependent terms in (2.1), calculating the current in this way is quite simple and
straightforward – considerably more simple than using the state observer described in §3.3.4 above.
Chapter 3
86
3.3.6. Feedback linearisation of discrete-time systems
A formulation of feedback linearisation also exists for discrete-time systems. Using this discrete-time
formulation is attractive because it is more practical to implement the compensator with a digital
processor. As explained above, previous applications of feedback linearisation to the loudspeaker
have used continuous-time formulations, and then used discretisations of certain parts in order to
implement the controller in a digital processor.
Using the discrete-time model of the loudspeaker developed in §2.3.4 above, it has been found that the
discrete-time formulation of feedback linearisation can be directly applied to a controller for
compensation of loudspeaker nonlinearity.
For the application of discrete-time feedback linearisation, it is necessary to consider the system in the
following state-space form
x[n + 1] = f (x[n]) + g(x[n])u[n]
y[n] = h(x[n])
(3.21)
where all these terms have the same meaning as for the continuous-time form of (3.1) above.
Application of feedback linearisation to the system of (3.21) by considering the system output at time
interval n + 1:
y[n + 1] = h(x[n + 1])
= h(f (x[n]) + g(x[n])u[ n])
(3.22)
If the derivative of the RHS of this with respect to the input u[n] is not zero, i.e. if
∂y[ n + 1]
≠0
∂u[ n]
(3.23)
then an input-output link is established, and the output can be solved in terms of the input. If this
derivative is zero, then it will be necessary to take the next output sample, as so
y[n + 2] = h(f (x[n + 1]) + g(x[n + 1])u[n + 1])
= h(f (h(f (x[ n]) + g(x[n])u[n])) + g(h(f (x[n]) + g(x[ n])u[n]))u[n + 1])
= h o (f , g u[n + 1]) o (f , g u[ n])
(3.24)
where o denotes ‘composition’. If the derivative of the RHS with respect to the input is again zero, it
will be necessary to take again a higher-order composition. This procedure is repeated until one finds
the derivative of the rth composition with respect to the input to not be zero, represented as
y[n + r ] = h r o (f , g u[n])
(3.25)
There is not a general expression for the inverse of this expression for u[n] as there is for the
continuous-time case in (3.7). The form of the solution will depend on the specific nature of the
system, input, and output vector fields, f, g, and h, respectively.
In the next section, this discrete-time formulation of feedback linearisation is applied to the discretetime model of a loudspeaker, developed in §2.3.4 above, to provide a simple algorithm for
compensation of nonlinear distortion in a loudspeaker.
Theory of active control of loudspeakers
87
3.3.7. Feedback linearisation with a discrete-time loudspeaker model
This section presents a new discrete-time implementation of feedback linearisation for compensation
of nonlinear distortion in a loudspeaker. This is done by applying the discrete-time formulation of
feedback linearisation presented above to the discrete-time nonlinear loudspeaker model presented in
§2.3.4
The motivation for developing a new algorithm for nonlinear distortion compensation is costreduction. In Chapter 5, discussions on the cost of nonlinear distortion compensation assume the only
cost of the compensation is the additional amplifier output it requires. An important consequence of
this is that the algorithm for performing the compensation must be as simple as existing algorithms
running on DSP’s in the target product – so as not to increase the cost of the hardware performing the
compensation processing. A rough quantification of this limit is that the algorithm should not be more
complicated than several second-order IIR filters. This limit precludes the use of more complex
distortion-compensation algorithms such as Volterra series methods, Neural Networks, NARMAX
models, or other ‘black-box’ methods. To this end, a new, simple, distortion compensation algorithm
suitable for DSP implementation has been developed and is presented here. As stated above, simplicity
has been maintained by using the nonlinear discrete-time model of the loudspeaker dynamics
developed in §2.3.4, and applying the discrete-time formulation of feedback linearisation.
This is an alternative to previous digital implementations of feedback linearisation for compensation
of nonlinear distortion. As discussed in §3.3.4, Schurer (1997) used numerical integration to simulate
a continuous-time model of the loudspeaker. The method presented here, using a discrete-time model
of the loudspeaker, avoids the need for this simulation.
Consider the displacement output from the voice-coil voltage by the nonlinear discrete-time model of
(2.85),


 1

{vc [n] − φ(x d [n]) u d [n]} − k1 (x d [n]) x d [n]
x d [n + 1] = σ x φ(x d [n]) 


 Reb

− a1 x d [n] − a 2 x d [n − 1]
(3.26)
where all the terms have the same definitions as in chapter 2. As per the theory of feedback
linearisation for a discrete-time system one considers the derivative of the output with respect to the
input. In (3.26), the output is xd[n], and the input is vc[n]. It can be seen by inspection of the RHS of
(3.26) that its derivative with respect to xd[n] will depend explicitly on the input vc[n].
According to the theory of feedback linearisation, the control law is obtained from (3.26) by inverting
the relationship defined between xd[n] and vc[n]. This inverted relationship, obtained by
straightforward algebraic manipulation, is as so:
v c [ n] =


 1

1
(x d [n + 1] + a1 x d [n] + a 2 x d [n − 1]) + k1 (x d [n])x d [n]  + φ(x d [n]) u d [n]
Reb 
 φ(x [n])  σ
d
 x



(3.27)
This equation may be interpreted as specifying the voltage needed to produce a certain displacement.
It may be used as a control law for nonlinear compensation, as it calculates the voice-coil voltage
needed to achieve a certain displacement, as per the nonlinear loudspeaker model of (3.26).
Interpreted in this way, it defines the input-output relationship of a controller, the input to which is the
specified displacement xd[n], and the output from which is the necessary voltage vc[n] to produce this
Chapter 3
88
displacement. By replacing xd[n] with rp[n] and vc[n] with rlin[n], this nonlinear control law may be
defined as
rlin [n − 1] =


σ φ(r [n − 1])
 r p [n] + σ x k1 r p [n − 1] + a1 r p [n − 1] + a 2 r p [n − 2] + x d
u d ·e [n − 1]
Reb
σ x φ(rd [n − 1) 

Reb
{
(
)
}
(3.28)
where rp[n] is the input to the nonlinear controller, rlin[n] is the output from the nonlinear controller,
and ud·e[n] is an estimate of the diaphragm-coil velocity. The estimate of the diaphragm-coil velocity
ud·e[n] is computed by differentiating the input to the nonlinear controller rp[n] as so:
u d ·e [n] = hdt [ n] ∗ r p [n]
= bdt ·0 r p [n] + bdt ·1 r p [n − 1] − a dt ·1u d ·e [n − 1]
(3.29)
where the coefficients of this differentiation approximation are as discussed in §2.3.4.
An important feature of the nonlinear control law of (3.28) is its simplicity. It effectively consists of a
second-order IIR filter, plus the addition of the polynomial evaluations for φ(rd [n]) and k1 (rd [n]) .
The studies presented in Chapter 5 consider that the nonuniformity in the suspension stiffness is
negligible. As shown in an example measurement in Figure 2.13, the stiffness nonuniformity is minor.
A necessary addition to the control law of (3.28) is the ‘re-introduction of linear dynamics,’ as per the
presentation of feedback linearisation of continuous-time systems presented in §3.3.1. For the case of
(3.28), this is done by pre-filtering rlin[n] with a filter having the linear response of the displacement
response. This can be done in a straightforward manner, with linear second-order IIR filter.
The effectiveness of a controller using the algorithm in (3.28) is evaluated by measurements presented
in §5.2.3, below.
One interesting development in the application of feedback linearisation to loudspeakers came from
Klippel (Jun. 1998). This paper showed that filtering for re-introduction of linear dynamics can be
combined with the nonlinear control law. Klippel’s work was done for the continuous-time
formulation of feedback linearisation. Attempts were made as part of research for this thesis to make
the same simplification to this discrete-time formulation of feedback linearisation, but were
unsuccessful. This may be a topic for further research.
3.4. Adaptive feedforward controllers
As discussed in the introduction, the general problem of a pure feedforward controller is its sensitivity
to model uncertainties. As discussed in §2.4, various characteristics of the loudspeaker change with
temperature and ageing. As the actual loudspeaker’s properties drift from those assumed by the
feedforward controller, the performance of the feedforward controller will decrease. This sensitivity
to model uncertainties was explained by Schurer (1997, p. 7) as the primary disadvantage of
feedforward nonlinear compensation systems.
Theory of active control of loudspeakers
89
As discussed in the introduction, tuning the parameters used in the model of the loudspeaker used by
the feedforward processor can be done by system identification.1 System identification is the process
of tuning the input-output characteristics of a model of a dynamic system to that of an actual dynamic
system. A feedforward controller using this features is generally referred to as an adaptive
feedforward controller.
Some literature has appeared on adaptive feedforward control for loudspeakers. Research on linear
adaptive feedforward control, focusing on loudspeaker-room equalisation, has been published by
Elliott and Nelson (1989), Kuriyama and Furukawa (1989), Radcliffe and Gogate (1992), Craven and
Gerzon (1992), and Elliott et al. (1994). These methods attempted to achieve equalisation of the
complete electroacoustic path, to the point of the listener, typically using high-order adaptive FIR
digital filters.
Research on nonlinear adaptive feedforward control has also been published. Klippel (Nov. 1998)
presented a method for parametric determination of the nonlinear characteristics of a loudspeaker by
nonlinear adaptive filtering.
In this thesis, linear adaptive control is of primary interest. It is considered that only the linear
properties of the loudspeaker are subject to drift.2 Although the advantages of nonlinear control are
the focus of the thesis, the nonlinear properties to be controlled are considered static – i.e. they can be
known a priori. This is because they are defined by the geometry of the construction of the
loudspeaker; it is assumed that this can be known for a given loudspeaker type.
Additionally, only parametric, or ‘grey box’ loudspeaker system identification is considered in this
thesis. High-order FIR filters, Volterra filters, NARMAX models, Neural networks and other ‘blackbox’ methods are not considered. This is a choice, driven by the need to keep the complete adaptive
feedforward controller simple. The need for simplicity is driven by cost.
Restricting loudspeaker system identification to parametric methods precludes using the FIR model
presented in §2.3.1, and requires the use of the IIR filter models §§2.3.2 and 2.3.3. The background
theory on adaptation of an IIR filter model is presented in the next section (§3.5). Chapter 4 presents
details of applying this theory to the loudspeaker, and measurements of its identification performance.
By a remarkable stroke of luck, the five parameters which vary with manufacturing tolerance,
temperature, and age, and thus cannot be known a priori, can all be determined from the electrical
impedance. Although six parameters define the complete lumped parameter model of the loudspeaker,
one of these – the diaphragm-coil moving mass md – can be known a priori, as it will not change
throughout the lifetime of the loudspeaker. This permits the feedforward processor to be fully tuned to
a loudspeaker by analysis of only the electrical impedance (i.e. without direct vibration measurement).
This is significant, as the electrical impedance can be analysed by a simple measurement of the
electrical voice-coil current, and does not need vibration measurement which is impractical and
expensive.
1
System identification is a general field in applied electrical, signal processing, control engineering, and basic
terms used for discussions on the subject are different in each of these three fields. The discussion used here
most closely follows that used in the field of signal processing.
2
As per the discussion in §2.4, the parameters known to drift with temperature and other considerations are
summarised in Table 2.1.
Chapter 3
90
3.5. System identification by adaptive filtering
As discussed in the introduction, an adaptive feedforward controller as shown in Figure 1.4 uses a
plant model to tune the feedforward controller to the plant. The plant model is implemented as an
adaptive filter performing system identification on the plant. This adaptive filter calculates a predicted
output yp[n] from the input u[n] using a weighting vector w, according to a filter function f (u[n], w ) .
The weighting vector w is tuned to minimise, in a mean-square sense, the error ε[n], calculated as the
difference between the measurable plant output ym[n] and the predicted plant output yp[n]. When the
adaptive filter has reduced the error ε[n] below some threshold, it will begin to update the feedforward
processor in Figure 1.4 accordingly.
In order to determine variation in those parameters given in Table 2.1, the filter function f (u[n], w )
must be designed along the specific dynamics of the plant. This is in contrast to many applications of
adaptive filtering, wherein the plant is treated as a ‘black-box’, the dynamics of which cannot be
known, and the purpose of the filter is simply to minimise the mean-square value of ε[n] in absolute
terms. It is particularly important that the dynamics of f (u[n], w ) be analogous to those of the plant if
an indirect output signal is measured such as the voice-coil-current. If in this case the dynamics of
f (u[n], w ) are different from the plant, the updated weight vector will not be readily usable by the
feedforward processor.
The difference between the measured plant output ym[n] and the predicted plant output provides an
error signal ε[n], as shown in Figure 3.9. The parameter vector w is determined by minimising the
error ε[n] in a mean-square sense, This is the basic arrangement for using an adaptive filter for system
identification, as described in standard textbooks by Widrow and Stearns (1985) and Haykin (1996).
y m [n ]
Plant Model
u[n]
f u[n], w
yp[n] -
+
Σ
ε[n]
Figure 3.9: Basic arrangement of an adaptive filter performing system identification.
3.5.1. General adaptive algorithms
Several methods exist for tuning the weight vector w such that it will minimise the error between the
filter’s predicted output yp[n] and the measured output ym[n]. Generally, such an optimally tuned filter
is referred to as a Wiener Filter.1
For audio systems considered in time-domain, the weighting vector is nearly invariably a real-valued
vector of length N, i.e.:2
w = [w1 w2 ...
wN ]
w ∈ ℜN .
(3.30)
An error surface ξ(w) is defined by the expectation value of the mean-square error, as so:
[
] [(
) ] = E [( y
ξ( w ) = E ε 2 [ n ] = E y m [ n ] − y p [ n ]
2
m [ n] −
f (u [n], w ))
2
]
(3.31)
1
The name ‘Wiener Filter’ comes from a definition for an optimally tuned continuous-time filter, published by
Wiener and Hopf (1931).
2
The weight vector w is defined here as a row-vector; in other presentations of adaptive filtering, it is typically
defined as a column-vector.
Theory of active control of loudspeakers
91
This error surface is the expectation value of the error for different values of the weighting vector w,
for a given input signal u[n] and a given measured signal ym[n]. The purpose of the adaptive filter is to
find the values of w which give the minimum value of ξ(w). This value of w is called the optimum
value, noted by wopt.
For an Nth-order linear FIR filter (also transversal, non-recursive, or purely feedforward filter), the filter
function f(u[n],w) is the scalar product between a vector u, a delay-line of the input signal of the same
length as w, and w as so:
f (u[n], w ) = u·w
= [u[n]
u[n − 1]
... u[n − N − 1] ]·w
(3.32)
In this case the error surface of Figure 3.9 becomes:
[
]
2
ξ(w ) = E ym [n] + w·Rw − 2p·w
(3.33)
where R is the input-signal autocorrelation matrix, and p is the input-signal-to-measured-signal crosscorrelation vector.1 It can be shown that if ym[n] is described by the signal u[n] by a linear, timeinvariant system, then the values of w which will minimise the error ε[n] in a mean-square sense are
given by2
w opt = R −1 p
(3.34)
This result in (3.34) is independent of the choice of the filter function f (u[n], w ) . However, it can
only be used if the matrix R can be inverted (is not singular). For the FIR filter, R will generally be
invertible, and thus (3.34) provides a general solution for finding wopt for this type of filter. However,
this is not generally used in practical adaptive filters, because it is computationally expensive to
calculate the matrix R and vector p and subsequently invert R. Considerable research has gone into
reducing the complexity of computing these values, leading to a body of algorithms referred to as
recursive least squares (or method of least squares, or sequential regression)3. These methods are not
presented nor is their application investigated in this thesis, as more simple algorithms have been
found to work, which are explained hereafter.
Instead of determining wopt in one step as in (3.34), adaptive filters usually use an iterative method. If
the error surface ξ(w) is uni-modal, i.e. is convex with a single global minimum, wopt may be
determined by updating w along the gradient of ξ(w). To this end, the gradient of ξ(w) is defined as
 ∂ξ( w )
d = ∇ξ ( w ) = 
 w0
∂ξ(w )
∂ξ(w ) 
...

w1
wN 
d ∈ ℜ N +1
(3.35)
Using the method of steepest descent, the weighting vector w can be iteratively shifted to its optimum
value. A new estimate of the weighting vector is obtained by subtracting from the its previous
estimate the gradient d, as scaled by convergence parameter µ4. In this way, one has w[n+1], an
updated estimate of w, obtained as so
1
These are standard concepts in adaptive filtering (signal processing). A basic explanation may be found on p.
20 of Widrow and Stearns (1985). A more detailed description may be found in §2.3 of Haykin (1996).
2
This represents the solution to the Wiener-Hopf equations for a linear, discrete-time system, where the weight
vector w describes the coefficients of an FIR filter; see p. 22 of Widrow and Stearns (1985) or p. 206 of Haykin
(1996).
3
See pp. 147-153 of Widrow and Stearns (1985), or pp. 483-533 of Haykin (1996).
4
The convergence parameter µ may also be a vector, in such case each element of the gradient vector d is
weighted differently.
Chapter 3
92
w[n + 1] = w[ n] − 12 µ d[ n]
(3.36)
where µ is the convergence parameter, and d[n] is the error surface gradient for weighting vector w[n].
The convergence parameter µ must be carefully chosen. If µ is too small, it will require large numbers
of iterations to obtain wopt; if µ is too large, the value w[n+1] will overshoot wopt, leading to erratic
convergence or unstable calculation.
The method of steepest decent does not itself provide a sufficiently simple algorithm for real-time
calculations. This is because the derivatives ∂ξ(w ) ∂wk are expensive to compute, particularly if they
must be recalculated at each time interval n. Widrow and Hoff (1960) found that the gradient of the
expectation value of the error, defined in (3.35), may be approximated by the gradient of the
instantaneous value of the error, as so:
∂y p [n]
∂ε[ n]
= −2 ε[n]
d wk [n] ≈ dˆ wk [n] = 2 ε[n]
∂wk
∂wk
(3.37)
where dˆwk [n] is the instantaneous estimate of the gradient of the error surface along the parameter wk
at time interval n. With this simplification, the weighting vector may be updated in the same manner
as (3.36) as so
w[n + 1] = w[n] − 12 µ dˆwk [ n]
= w[n] + µε[ n]∇ w y p [n]
(3.38)
where ∇ w y p [n] is a vector containing the instantaneous values of the derivatives of the output with
respect to each element of the weighting vector w at time index n. This method for updating w was
named by Widrow and Hoff as the LMS algorithm. It is referred to as a stochastic gradient method,
due to its estimation of the true gradient, defined in (3.35), with only its instantaneous value.
Additionally, one does not need to consider the derivative of the square of the error with respect to
each weighting coefficient, but simply the derivative of the error signal itself. Experience has shown
that this estimation leads only to a random error in the estimate of the gradient, which averages to zero
over multiple iterations of updating, leading to an unbiased estimate of wopt. The remarkable
combination of simplicity and effectiveness of this algorithm have led to its widespread commercial
use in adaptive equalisation, control, array beam-forming, and echo cancellation, among other
applications.
The most common filter structure used for adaptive filtering is the FIR filter, as it is inherently stable.
For this type of filter, the LMS algorithm reduces to a form even simpler than (3.38). The derivative
∂y p ∂wk reduces to u[n − k − 1] , producing
w[n + 1] = w[n] + 2µε[n]u[n]
(3.39)
3.5.2. Adaptive IIR filters
As explained in §2.3 above, a compact-discrete-time model of a loudspeaker requires an IIR filter. It is
known that adaptive algorithms for IIR filters have several difficulties that those for FIR filters,
described above, do not. Recent tutorials by Shynk (1989) and Netto et. al. (1995) describe these
problems, and how different adaptive algorithms addressed them. Briefly, these problems are:
• Risk of instability
• Slow convergence rate
• Risk of convergence to local (non-global) error minima
Theory of active control of loudspeakers
93
These problems have been judged to be surmountable for the discrete-time loudspeaker model for two
reasons:
• A great deal of a priori knowledge of the ‘system-to-be-identified,’ the electrical admittance of the
loudspeaker, is available.
• Initial guess of the filter’s parameters can be made within a short range from their actual value.
The LMS IIR algorithm, is identical to the LMS algorithm for the FIR algorithm of (3.39), except for the
expression for the derivatives of output error with respect to the weighting coefficients. For the IIR
filter, the filter output is calculated recursively, complicating the definition of the derivative of the
output with respect to the weighting coefficients wk. Consider the basic definition of an IIR filter,
wherein the filter output is calculated as:
y p [n] = b0 x[n] + b1 x[n − 1] + ... + bM x[n − M ] − a1 y p [n − 1] − a 2 y p [n − 2] − ... − a N y p [n − N ] (3.40)
According to the ‘small step-size approximation’ developed by White (1975), the derivatives of the
output yp[n] with respect to the feedback weighting coefficients ak and feedforward coefficients bk may
be approximated by the following recursive calculation
∂y p [n]
∂ak [n]
∂y p [n]
∂bk [n]
≈ y[n − k ] +
N
∑ a [ n]
k
n =1
≈ x[n − k ] +
M
∑ a [ n]
k
n =1
∂y p [n − k ]
∂ak [n]
∂y p [n − k ]
∂ak [n]
=&
α k [ n]
(3.41)
=&
βk [ n]
(3.42)
With this definition, the derivatives may be calculated recursively as so
α k [n] = y[n − k ] +
N
∑ a [ n]
k
k =1
βk [n] = x[n − k ] +
M
∑ a [ n]
k
k =1
∂y p [n − k ]
∂ak [ n]
∂y p [n − k ]
∂ak [n]
(3.43)
(3.44)
This is an important general feature of the LMS algorithm for IIR filters. It will be applied to various
filter structures serving as discrete-time models of the loudspeaker. The details of this application is
the subject of the next chapter.
It has been proposed by Feintuch (1976) that the latter sums in (3.41) and (3.42) can be discarded.
However, it remains contentious discarding these sums does or does not lead to bias error in the
converged values. For this reason, this simplification has not been used. It may be a suitable subject
for further research.
Adaptive lattice-form IIR filters have received much interest in recent years (Parikh et al., 1980). The
primary advantage of lattice-form IIR filters is the simplicity with which their stability may be
assessed. However, it was found as part of background research for this thesis that the connection
between physical parameters of a system and the parameters of a lattice filter are considerably more
complicated than in the direct IIR form presented above. For this reason, lattice-from IIR filters are not
investigated for the loudspeaker system identification.
94
Chapter 3
3.6. References
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Theory of active control of loudspeakers
95
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Hall, David S., “Design Considerations for an Accelerometer-Based Dynamic Loudspeaker Motional
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98
Chapter 3
99
4.
Loudspeaker system identification
This chapter presents methods for loudspeaker system identification. The focus is specifically on
those aspects of the loudspeaker which cannot be known outright from its design and manufacturing
specifications. These are properties of the loudspeaker which vary due to manufacturing tolerances,
temperature changes, and ageing, as discussed in §2.4.
In order to ensure that the algorithms for system identification meet the criterion for simplicity set out
in the introduction, i.e. that the algorithms not be more complicated than existing audio DSP
algorithms, an iterative system identification procedure must be used, using adaptive filtering, the
basics of which were presented in §3.5, above.
Again, to simplify the system for active control, an adaptive filter structure is chosen which
corresponds to the physical dynamics of the loudspeaker. As discussed in control systems theory, this
is to say the system identification uses grey-box model structures as opposed to black-box model
structure (Ljung, 1999, p.13). In this way, the parameters identified by the adaptive filter can be
directly used by the feedforward processor, and do not require transformation (as would be necessary
if a black-box model were used1). It is for this reason that the discrete-time model for a loudspeaker is
developed in §2.3, above.
An overview of the approach to system identification is given in §4.1. This section first describes the
three different error signals used. A description of the hardware implementation and the loudspeakerunder-test is given in §4.1.2. A description of the software implementation of the algorithms is given
in §4.1.3.
The approach to loudspeaker system identification is nearly identical to that described by Knudsen et
al. (1989). The parameter updating algorithm used here, however, is the standard LMS IIR output-error
algorithm (presented in §3.5.2). Knudsen et al. used techniques more common to off-line system
identification, which can be roughly understood in the terms of adaptive signal processing as batchprocessing equation-error or ARMAX techniques. These latter parameter updating methods are by far
more stable and robust than the LMS IIR output-error algorithm. However, they were considered too
computationally expensive for the present application. As part of research for this thesis, it was found
that by using special a priori knowledge available for a given loudspeaker, convergence, stability and
robustness of the IIR LMS algorithm could generally be guaranteed. Discussions on how this was done
are presented in §§4.1.4 - 4.1.6.
4.1. Overview of approach, implementation, and evaluation
A brief overview of the three different forms for generating an error equation under study in this thesis
are presented in §4.1.1. The hardware
The linear dynamics of the loudspeaker in all of these forms are based on second-order IIR filters. As
the parameters of these IIR filters are adapted, the conditions for stability of an IIR filter are reviewed
1
One example of a black-box model for the loudspeaker would be an FIR model of the electrical impedance. As
discussed in §2.3.1, it would be necessary to identify the physical parameters of a loudspeaker from the
coefficients of the adapted FIR filter before they could be used by a feedforward processor. This type of
processing is considered too computationally expensive.
Chapter 4
100
in §4.1.4. In early trials of the system identification algorithms, it was found that the convergence was
either slow or erratic.
It was found that, by using tolerance information which can be known a priori about any loudspeaker,
the parameters of the IIR filters could be kept within a minimum distance of their optimum values.
This has been achieved by use of a ‘tolerance quadrilateral,’ presented in §4.1.5.
4.1.1. Forms for developing an error equation
As per the discussion on feedback processing for loudspeakers in §3.1, direct output signals from the
loudspeaker such as a sound field measurement or vibration measurement are expensive and/or
impractical to obtain. For this reason, plant models (loudspeaker models) defining the structure of the
adaptive filter for system identification are considered only for the electrical characteristics of the
loudspeaker. These consider (in discrete time) measurement of the voice-coil voltage vc·m[n] and the
voice-coil current ic·m[n]. Two obvious plant models suggested by Knudsen et al. (1989) with these
two signals available are:
• Electrical admittance (prediction of current from the measured, or known, voltage)
• Electrical impedance (prediction of voltage from the measured current)
In the electrical admittance output error form, an adaptive filter makes an estimate (‘prediction’) of
the voice coil current ic·p[n]. The input to this filter is the measured voice-coil voltage vc·m[n] and
measured voice-coil current ic·m[n]. In this form, the electrical inductance Leb is assumed to be
negligible. With this assumption, the basic voltage equation of (2.1) has no differential operators, and
can thus be expressed directly in discrete time as so:
v c [n] = Reb ic [n] + φ( x d [n]) u d [n]
(4.1)
Solving for current in this equation produces
i c [ n] =
1
(vc [n] − φ(x d [n]) u d [n])
Reb
(4.2)
The strategy of this ‘admittance output error form’ is to predict the electrical current with this
equation, using a measurement of the voice-coil voltage, and a prediction of the diaphragm-coil
velocity. The classical adaptive filtering techniques described in §3.5 are used to adapt free
parameters in order to minimise the difference between the measured and predicted current, in a meansquare sense. Investigations into the performance of this plant model structure for loudspeaker
parameter identification are presented in §4.2, below.
In the electrical impedance output error plant model structure, the measured voice-coil current ic·m[n]
is used to calculate a predicted voice-coil voltage vc·p[n]. This is done directly as per the voltage
equation of (4.1). This predicted voice-coil voltage is compared with the measured voice-coil voltage
vc·m[n], and as per the electrical admittance structure described above, parameters are tuned to
minimise the error between the two. (Note that since solid-state power amplifiers provide constantoutput voltage, the voice-coil voltage can be known from the output signal from the controller, and
thus need not be measured separately.) It is shown that identification of the electrical inductance Leb
can also be performed with only a modest increase in complexity. Investigations into the performance
of this plant model structure for loudspeaker parameter identification are presented in §4.3, below.
An alternative to these two plant model structures are motional-signal equation error techniques. Such
a technique was presented by Klippel (1999), wherein the diaphragm-coil velocity ud(t) is predicted by
the force equation and the voltage equation, both including parametric nonuniformity of the type
described in §2.2.1. An error signal, to be minimised in a mean-square sense, is obtained from the
difference between the velocity predicted by the force and voltage equation, as so:
Loudspeaker system identification
ε eeu [n] =

1
 v c·m [n] − Reb ic·m [n] − Leb hdt [n] ∗ ic·m [n]
φ(x d [n]) 
101



 − h Y m [n] ∗  φ(x d [n]) ic [ n] 



(4.3)
where hdt[n] is the impulse response of a differentiation approximation as discussed in §2.3.4, and
h Y m [n] is the impulse response of the mechanical mobility as discussed in §2.3.3.
Klippel showed how this method could be used to identify coefficients of truncated polynomial series
approximations to the parametric nonuniformity in the transduction coefficient, suspension stiffness,
and blocked electrical inductance.
One disadvantage in the development of this error equation in (4.3) is that it requires the displacement
signal xd[n]. This is needed for evaluation of the transduction coefficient nonlinearity, i.e. evaluation
of φ(x d [n]) . This can, of course, be obtained by integration of the displacement signal. However, it
has been found to be simpler to base the error equation on the displacement, obviating the need to
compute it as an extra step. This form is based on this error equation:
 1

 v c·m [n] − Reb ic·m [n] − Leb hdt [n] ∗ ic·m [n]
ε eex [n] = hdt [n] ∗ 
(
)
[
]
φ
x
n

d




 − h X m [n] ∗  φ(x d [n]) ic [n] 



(4.4)
where h X m [n] is the impulse response of the mechanical receptance, as discussed in §2.3.2. This
method is referred to here in as the displacement equation error plant model. Basics of its
performance are presented in §4.4. It is found that this displacement equation error method is
considerably more complicated than the current and voltage output error methods described above. For
this reason, this method is not considered in complete detail.
The system identification techniques developed in this chapter are suitable only for a loudspeaker
mounted in a closed box. Modelling more complex acoustic enclosures has not been considered.
4.1.2. Hardware implementation and system-under-test
The adaptive algorithms for system identification were investigated on actual loudspeakers. A block
diagram of the hardware system used is shown in Figure 4.1. Note that Figure 4.1 shows a block for
the feedforward processing systems as well, although these are not discussed in this chapter.
Chapter 4
102
Feedforward processing
w [n]
Linear
Dynamics
xm·target [n]
Plant
Power
Amplifier
Inverse
Dynamics
D/A
anorm
Parameter
update
vc·m [n]
ic·m[n]
Rlead
Rshunt
Rlead
Loudspeaker
+
-
System
Identification
adaptive
filter
ic·p[n] -
Σ
+ ic·m[n]
εoei[n]
sic
svc
A/D
A/D
gic
gvc
Figure 4.1: Block diagram of hardware used for investigation of the loudspeaker system identification
algorithms presented in this chapter. Note this figure also shows the ‘feedforward processing’ blocks,
presented in §3.3, and investigated experimentally in chapter 5. This chapter presents theory and
experimental results of only the ‘system identification’ block.
A photograph of the mounting of the loudspeaker under-test for experimental trials of the system
identification algorithms presented in this chapter is shown in Figure 4.2. A detail of this figure is
shown in Figure 4.3. The loudspeaker used is a standard ø16mm microspeaker currently available on
the market. This is a widely used microspeaker, for reproducing speech and alert-tones in handportable phones. Its general features and proportions are as shown in Figure 2.1.
Loudspeaker system identification
Variable
back-cavity
volume
Scanning mirrors
103
Laser vibrometer
Sensor head
Piston handle
Mounting brace
Figure 4.2: Photograph of mounting of loudspeaker under-test during experimental trials of the systemidentification algorithms presented in this chapter.
Variable
back-cavity jig
ø16mm
Loudspeaker front
Figure 4.3: The loudspeaker is highlighted in the photograph by the broken white ellipse. The red dot at the
centre of the speaker is reflected light from the laser vibrometer, used to verify results of the system
identification.
The red1 dot visible at the centre of the loudspeaker Figure 4.3 is reflected light from the laser
vibrometer. The laser vibrometer was used to verify the values to which the system identification
algorithms converged. Signals from the laser vibrometer were not used by the system identification
algorithms directly. As per Figure 4.1, the algorithms used only electrical voltage and current signals.
1
The dot will of course appear white if a black-and-white printing process was used.
104
Chapter 4
The loudspeaker was mounted in a variable back-cavity jig, the stainless-steel and aluminium
cylindrical structure shown in Figure 4.2. This provides an acoustically sealed back-cavity for the
loudspeaker such that it may be modelled as described in §2.1.7. The total volume of the rear-cavity
may be easily varied via a travelling piston. A threaded handle for the piston, as labelled in Figure
4.2, permits course and fine adjustment of the rear-cavity volume. This permits direct variation of the
effective resonance frequency f0. This proved considerably useful for quick analysis of the tracking
performance of the system identification algorithms.
Additional photographs of the experimental set-up for evaluation of the system identification
algorithms are presented in Appendix A.
4.1.3. Software implementation
The system identification algorithms were implemented on a standard desktop PC, using an Intel
Pentium II processor, with a clock speed of 266 MHz. The algorithm was written using standard ANSI
C functions. Analogue input/output was facilitated with a standard sound card. A header file written
by Antti Vähätalo of Nokia Research Center was used to handle data transfer between the C-program
and the sound card. No dedicated audio DSP processor was used. Despite not using any special,
dedicated audio DSP processing hardware, no problems with real time performance were found, even
at the highest sampling rate of 48 kHz.1
No special programming tricks were used in the implementation of the algorithms presented herein.
Programming was done more or less by direct implementation of the difference equations written in
the thesis. For this reason, the C-code used for processing is not presented here, as it does not contain
any new information from that which is already presented in the equations.
The operation of the algorithm was controlled by a simple graphical user interface. A snapshot of this
graphical user interface is shown in Figure 4.4.
1
Your author was surprised to not have a problem with real-time performance at this highest sampling rate of
48kHz. It had generally been assumed that PC’s were not capable of running any type of audio DSP algorithms
on their own, i.e. without using dedicated audio DSP processors. That these algorithms were able to run in real
time without the use of dedicated DSP hardware served as a testament to your author to the advances made in
the processing capabilities or ordinary desktop PC’s in recent years.
Loudspeaker system identification
105
Figure 4.4: User interface for C-programs. This interface has controls for the system identification algorithm
presented in this chapter, as well as for the nonlinear distortion compensation algorithm presented in §3.3.7
and investigated experimentally in §5.2. The version of the interface shown in this figure is for controlling the
electrical-current output-error form of the system identification algorithm, presented in §4.2.
Data variables and computation used double floating point precision. Problems associated with fixedpoint computation were not considered.
Sampling rates of 8 kHz, 16 kHz, and 48 kHz were investigated. Convergence time of the algorithm,
as measured in seconds, is not appreciably different for the different sample rates. All of the
convergence plots presented below were recorded from data processed at 16kHz.
Chapter 4
106
4.1.4. Stability triangle
According to elementary theory of discrete-time signal processing (Oppenheim and Schafer, 1989), an
IIR filter is inherently stable only if the poles of its z-domain transfer function all lie inside of the unitcircle of the z-plane. For a second-order IIR filter, this criteria may be expressed as three inequalities
which must all be true:
a2 < 1
a2 > (− a1 − 1)
(4.5)
a2 > (a1 − 1)
These criteria are plotted graphically in Figure 4.5
a2
a2 < 1
1
a 2 > − a1 − 1
-2
-1
0
2
a1
a 2 > a1 − 1
Figure 4.5: Geometric representation of the stability criteria described in (4.5)
The stability triangle was used in early trials of the adaptive IIR algorithm, by assessing the
truthfulness of the inequalities in (4.5). If any of these inequalities were found to be untrue, the
parameters were not updated by the instantaneous gradient estimate.
Although use of the stability triangle in this way did prevent unstable calculation, convergence was
generally poor. As discussed below, this was found to be due to the nature of the error surface in ak,
and not due to instability.
4.1.5. Tolerance quadrilateral
In early trials of the electrical-current output-error algorithm, convergence of the ak parameters was,
erratic, slow, and/or unpredictable. For some initial values, the parameters would converge rapidly.
Other initial values lead to erratic updating, eventually forcing the ak values to the edge of the stability
triangle. Still other trials showed initial convergence of the parameters toward their optimal values,
resulting in unstable convergence as they approached the optimum values.
In order to better understand the convergence dynamics, a simulation was made of the a1, a2 error
surface. The error was calculated over a range of values of a1, a2 for 4 seconds of measured data on an
actual loudspeaker. The resulting error surface is shown in a contour plot in Figure 4.6, and in a 3-D
surface mesh in Figure 4.7. Note that the parameter space region over –1 < a2 < 0 is not analysed.
Although this region is within the stability triangle, it does not correspond to physically realisable
parameter values, as will be shown below.
Loudspeaker system identification
107
It is particularly clear from Figure 4.7 that this error surface does have a single global minimum,
occurring at around a1 = –1.8, a2 = 0.97. Thus it does not suffer from the problem of local minima of
some adaptive IIR filtering cases as described by Johnson and Larimore (1977).
The problematic feature of this error surface is that its gradient is sharp only in the vicinity of the
global minimum. As can be seen from the contour plot (Figure 4.6), for much of the a1, a2 parameter
space the error is around 26~27%, with a very small slope. As, for the LMS algorithm, the
convergence rate is directly proportional to the slope, selecting initial values in this region will lead to
very slow convergence. Although a higher convergence parameter could be used to increase the
convergence rate, this would lead to large parameter spread and potentially unstable convergence as
the parameters approach the global minimum, where the slope of the error surface is larger.
The solution to this problem has been to use a priori information available about the actual variation
in a1 and a2. It was explained in §2.4 above that the loudspeaker’s parameters can be known within
certain tolerances. By translating these known tolerances in the physical parameters into a1 and a2, the
region of the a1, a2 parameter space which must be considered by the adaptive algorithm may be
reduced to that region of the space wherein the slope of the error surface is not small.
Chapter 4
108
100
0.9
90
0.8
80
0.7
70
0.6
a
2
60
0.5
50
0.4
40
0.3
30
0.2
20
0.1
0
−1.5
−1
−0.5
0
a
0.5
1
1.5
10
1
Figure 4.6: Contour lines of the error surface, in percent. The difference in error between each contour line is
0.1%.
Figure 4.7: Plot of the error surface vs. a1 and a2.
Loudspeaker system identification
109
It is first necessary to consider how variations in the resonance frequency (normalised to Fs) and
damping ratio ζ translate to changes in a1 and a2. In Figure 4.8, lines of constant resonance frequency
and damping ratio are shown in the a1, a2 parameter space. As can be seen from this figure, in the
upper left-hand corner, where the minimum in the error surface occurs, these lines are approximately
straight. That these lines are straight permits imposition of frequency and damping tolerance criteria
by simpler expressions in a1 and a2 than the direct formulae of (2.66).
0.45Fs/2
1
0.9
ing
Increasing Damp
0.8
0.7
0.6
Fs/4 0.55Fs/2
ζ = 5%
Increasin
g Freque
ncy
ζ = 10%
ζ = 15%
ζ = 20%
a2
F /2
s
0.5
0.4
0.3
0.2
0.1
0
−2
ζ = 500%
−1.5
−1
−0.5
0
0.5
1
1.5
2
a1
Figure 4.8: Lines of constant resonance frequency (normalised to Fs) and damping ratio ζ in the a1, a2
parameter space. Notice no lines go below a2 = 0, even though this would result in a stable filter.
Imposition of known limits in variations in the resonance frequency and damping ratio restricts the a1
a2 parameters space which must be searched. By considering the following limits:
f act = f 0 ± 25%
ζ act = ζ 0 ( − 90% + 360% )
the a1, a2 space is reduced to the region shown in Figure 4.9.
(4.6)
Chapter 4
110
1
1
ζmin
0.9
0.98
0.8
0.94
fmin
fmax
a2
0.96
0.6
a2
0.7
0.5
0.92
0.4
0.9
ζ
0.3
max
0.88
0.2
0.86
0.1
0
−2
−1.5
−1
−0.5
0
a
0.5
1
1.5
2
1
0.84
−1.9
−1.8
a
−1.7
1
Figure 4.9: Normalised resonance frequency and damping ratio tolerances of (4.6) in the [a1 ,a2 ]parameter
space. [left] complete region of physically realisable [a1, a2] values; [right] focused view around target
values. The ‘×’ in both graphs indicates the target value.
As discussed above and can be seen from Figure 4.9, the limits form approximately straight lines in a1,
a2. This permits maintaining updated values of a1 and a2 within these limits to be performed by this
algorithm:
if (a 2 > ζ min·1 a1 + ζ min·0 ) , a 2 = ζ min·1 a1 + ζ min·0
if (a 2 < ζ max·1 a1 + ζ max·0 ) , a 2 = ζ max·1 a1 + ζ max·0
if (a1 > f max·1 a 2 + f max·0 ) , a1 = f max·1 a 2 + f max·0
if (a1 < f min·1 a 2 + f min·0 ) ,
(4.7)
a1 = f min·1 a 2 + f min·0
where the terms [ζmin·1 , … , fmin·0 ] are coefficients defining the four lines of a quadrilateral
approximating the limits in Figure 4.9.
4.1.6. Frame-based updating
Most audio signal processing uses frame-based processing. The motivation behind this comes
primarily from lossy-compression (coding) techniques, referred to as speech or audio coding. These
algorithms operate in the frequency domain, and therefore must operate on one frequency-transformed
frame of time data at a time. Typical frame lengths are on the order of 20ms, though this can vary
significantly depending on the application.
Such frame-based processing has no direct theoretical impact on the adaptive algorithm developed
here. However, it can be used to reduce the computational overhead required for assessment of the
tolerance quadrilateral. It has been found that the estimate of the gradient can be ‘accumulated’ over
one frame such that
dˆ ak ·acc [ j ] =
N frame
∑ dˆ
n =1
ak
[ n]
(4.8)
where dˆ ak ·acc [ j ] is the gradient estimate accumulated over the jth –frame, and where the frame has
Nframe samples. The adapted parameters are then updated according to this accumulated gradient
according to
ak [ j + 1] = ak [ j ] − µ ak dˆa k ·acc [ j ]
(4.9)
Loudspeaker system identification
111
The advantage of this technique is that the tolerance quadrilateral assessment of (4.7) need be made
only once for each frame, instead of for each sample. Although the parameters are updated only once
per frame (instead of for each sample), due to the stochastic nature of estimation of the gradient in the
LMS algorithm, the accumulation of the instantaneous estimate of the gradient over the frame length
results in a more accurate estimate of the gradient for the end-of-frame updating. The net results is
that the convergence rate using only end-of-frame updating tends to be about the same as sample-bysample updating.
Chapter 4
112
4.2. Electrical current output-error form
The adaptive filter for the electrical current output error plant model structure is shown in Figure 4.10.
The method for developing this model is the same as used to develop the discrete-time nonlinear
loudspeaker model presented in §2.3.4.
vc·m[n]
φ(xd)
ic·m[n]
φ(xd )
+
Σ
fc·p[n]
-
Linear mechanical
model
+
ud[n]
xd[n]
-
Σ
1/Reb
ic·p [n]
-
Σ
+ ic·m[n]
εoei[n]
k1(xd)
Figure 4.10: Block diagram of electrical current output error (ECOE) plant model structure. Notice
that linear and nonlinear components are separated. This diagram shows the inclusion of nonuniform
force factor and stiffness. Other parameter nonuniformity, as discussed in §2.2.1, may be included in
a similar manner, though in some cases require more complicated linear dynamics models, e.g.
calculation of acceleration.
This plant model calculates a predicted current signal ic·p[n]. An error signal εoei[n] is obtained by
comparing the predicted current signal with a measured current signal ic·m[n] as so:
εoei [n] = ic·m [n] − ic· p [n]
(4.10)
The predicted current signal is calculated according to the voltage equation of (2.1). By ignoring the
effects of the blocked electrical inductance Leb (as discussed towards the end of §2.1.1), the predicted
current is calculated as per:
i c · p [ n] =
1
(vc·m [n] − φ(x d [n]) u d [n])
Reb
(4.11)
where vc·m[n] is the measured voltage, ud[n], is the diaphragm velocity predicted according to the force
equation, and other terms are as previously presented. The diaphragm velocity ud[n] and displacement
xd[n]are computed according to the ‘linear mechanical model’ in Figure 4.10, incorporating the linear
second-order dynamics of the mechanical dynamics of a loudspeaker mounted in a closed box, as
discussed in §2.1.7. These second-order dynamics are implemented in discrete time using the
discrete-time models of mobility (presented in §2.3.3) and receptance (presented in §2.3.2) to calculate
the velocity ud[n] and displacement xd[n], respectively, from the force on the voice-coil fc·p[n].
Explicitly, they are calculated as per this second-order IIR filter:
ud [n] = σu f c· p [n] − σu f c· p [n − 2] − a1ud [ n − 1] − a2ud [n − 2]
(4.12)
Loudspeaker system identification
113
where σu is the reference sensitivity for the discrete-time mobility z-domain transfer function defined
in §2.3.3, a1 and a2 are the feedback coefficients defining the poles of the system as per §2.3.3,1 and
fc·p[n] is the predicted force on the voice-coil predicted from the electrical current as per:
f c· p [n] = φ(xd [n])ic·m [n] − k1 (xd [n]) xd [n]
(4.13)
The displacement signal is calculated iteratively from the force using
xd [ n] = σ x f c· p [n − 1] − a1 xd [n − 1] − a2 xd [n − 2]
(4.14)
as defined in §2.3.2, and fc·p[n] is the predicted force as per (4.13).
Notice that the displacement signal xd[n] appears in the expression for the predicted force fc·p[n] and
vice-versa. Thus it would at first appear that this equation pair form only an implicit and not explicit
method for calculating the displacement from the current. However, notice that the force fc·p[n]
appears as a delayed sample in the equation for xd[n]. It is thus possible to ‘predict’ the displacement
one-sample ahead. This can then be used in the calculation of the next sample of the predicted
force fc·p[n].
Linear case
The linear case, wherein parameter nonuniformity with respect to displacement is assumed to be
negligible, is first considered. If models of the parameter nonuniformity are known, this linear case
may be defined by the approximations
φ(x d [n]) ≈ φ 0
(4.15)
k1 (x d [n]) ≈ 0
Given the above assumption, the formula for calculating predicted force fc·p[n] in (4.13) simplifies to
f c· p [n] = φ0ic·m [n]
(4.16)
Combining (4.11) and (4.12) and assuming the linear case described above, the error is calculated by
εoei [n] = ic·m [n] −




1 
vc·m [n] − φ0  φ0σu  ic·m [n] − ic·m [n − 2] − a1ud [n − 1] − a2ud [ n − 2]
Reb 




(4.17)
where ud[n] is calculated iteratively as per (4.12), and where fc·p[n] is calculated from the measured
current ic·m[n] as per (4.16).
The LMS algorithm is used to iteratively update the parameters Reb, φ0, a1 and a2 such that the error
signal εoei[n] is minimised in a mean-square sense. As explained in §3.5.1, the LMS algorithm uses a
simple stochastic gradient, determined from the product of the error signal and the derivative of the
error signal of the parameter to be updated. As the error signal defined in (4.17) is not of standard FIR
or IIR form, its derivatives for each of the parameters to be updated are derived for each parameter in a
separate section below.
4.2.1. Updating Reb
The derivative of the error, defined for the linear case in (4.17), with respect to the DC-resistance Reb is
given by
1
Note that these feedback coefficients are the same as those for the mechanical displacement. This is because
the mechanical mobility has the same pole locations as the mechanical receptance.
Chapter 4
114
1
1 ∂
∂
(vc·m [n] − φ0 ud [n])
εoei [n] = 2 (vc·m [n] − φ0 ud [ n]) −
Reb ∂Reb
∂Reb
Reb
(4.18)
Given that vc·m[n] is a measured value, and that ud [n] is calculated from the measured value ic·m[n]
without use of Reb, the derivative with respect to Reb is zero. Therefore, the last term in this equation is
zero, reducing the partial derivative of the error εeoi[n] with respect to the DC resistance Reb to
∂
1
εoei [ n] = 2 (vc [n] − φ0 ud [n])
∂Reb
Reb
(4.19)
Notice that the term in parentheses on the RHS of (4.19) is the same as that appearing in the definition
of the predicted current ic·p[n]. Thus this derivative may be written more compactly as
ic· p [n]
∂
εoei [n] =
∂Reb
Reb
(4.20)
From this the instantaneous estimate of the gradient of the error surface along Reb is given by
∂ε [n]
dˆReb [n] = εoei [n] eoi
∂Reb
= εoei [n]
(4.21)
ic· p [n]
Reb
with which the estimate of the DC-resistance may be updated using the LMS algorithms as so
Reb [ n + 1] = Reb [n] − µ Reb εoei [ n]
ic· p [ n]
Reb
(4.22)
4.2.2. Updating ak
As noted above, neither the measured current ic·m[n] nor the predicted force fc·p[n] depend on the
feedback coefficients ak. Therefore the derivative of the error with respect to ak is that of a standard
output-error IIR algorithm where the IIR filter concerned is that for the mechanical mobility. Thus
generally one has
∂
∂
εeoi [n] =
− ic· p [n]
∂ak
∂ak
(
=−
=
)
∂ 1
[vc·m [n] − φ0ud [n]]
∂ak Reb
(4.23)
φ0 ∂
ud [ n]
Reb ∂ak
The derivative of the velocity predicted from the force equation ud[n] with respect to the feedback
coefficients ak is
∂
∂u [ n − 1]
∂u [n − 2]
− a2 d
ud [ n] = −ud [n − k ] − a1 d
.
∂ak
∂ak
∂ak
(4.24)
By defining
α k [ n] =
∂
ud [ n] ,
∂ak
(4.25)
Loudspeaker system identification
115
the derivative of ud[n] with respect to the feedback coefficients ak may be calculated iteratively
according to
α k [ n] = −ud [n − k ] − a1α k [n − 1] − a2α k [n − 2] .
(4.26)
The instantaneous estimate of the gradient of the error surface along the parameter ak is therefore
given by
∂ε [n]
dˆ ak [n] = ε[n] eoi
∂a k
(4.27)
φ
= ε[n] 0 α k [ n]
R eb
where αk[n] is calculated iteratively as per (4.26). The ak parameters are updated according to the LMS
algorithm as so:
ak [n + 1] = ak [ n] − µ a k dˆa k [n]
(4.28)
where µ ak is a convergence parameter specific to ak. Summing up explicitly, this is
ak [n + 1] = ak [n] − µ a k ε[n]
φ0
α k [ n]
Reb
(4.29)
4.2.3. Updating of σu as a feedforward coefficient
The partial derivative of the error with respect to the overall gain of the mobility is given by
 φ ∂
∂
∂
∂   1
φ
vc [n] − 0 ud [n]  = 0
ud [ n]
ε eoi [n] =
− ic· p [ n] =
−


Reb
∂σu
∂σu
∂σu   Reb
  Reb ∂σu
(
)
(4.30)
The partial derivative of the velocity ud[n] with respect to the overall gain σ0 is
∂u d · f [n − 1]
∂u d · f [n − 2]
∂
u d · f [n] = f c· p [ n] − f c· p [n − 2] − a1
− a2
∂σ u
∂σ u
∂σ u
(4.31)
The partial derivative of the velocity ud[n] with respect to σ0 may be defined as β σ 0 [n] , and calculated
recursively as so
β σu [n] = f c· p [n] − f c· p [n − 2] − a1β σu [ n − 1] − a 2 β σu [n − 2]
(4.32)
The term σ0 is then updated in the traditional manner as so
σ u [n + 1] = σ u [n] − µ σu ε[n]
φ0
β σ [ n]
Reb u
(4.33)
Attempting to update σu in this manner leads to poor convergence of the parameters in general. This is
understood to be a reflection of the fact that σu is not an independent parameter; it is determined
directly by a2, as explained in §2.3.2, above. For this reason, a different approach to updating this
parameter is taken, which is explained in the next section.
4.2.4. Updating of σu as a dependent variable, defined by a2
It is shown in §2.3.2 that σu can be accurately approximated as a third-order polynomial function of a2
as so
Chapter 4
116
σ u (a 2 ) =
1
mt Fs
3
∑p
n =0
n
σu ·n a 2
(4.34)
Thus σu may be determined from updated values of a2, instead of being updated as an adaptive filter
coefficient. Thus its derivtive with respect to the error εoei[n] is not considered.
However, it is important to note that caclualting σu as per (4.34) changes the derivative of the error
with respect to a2, because σu becomes a function of a2. To consider how to re-write the derivative of
the error εoei[n] with respect a2, the derivative of σu is first considered. With σu defined as (4.34), its
partial derivative w.r.t a2 is defined as, and calculated by
∂ a2 σ u (a 2 ) =&
∂σ u
∂a 2
1
=
mt Fs
N −1
∑
n =0
(4.35)
np σu ·n +1 a 2n −1
With this definition and method for calculation of the derivative of σu with respect to a2, the derivative
of the error εoei[n] with respect to a2, originally given in (4.26), becomes
∂
∂
∂u [n − 1]
∂u [n − 2]
εeoi [ n] =
ud [n] = ∂ a 2 σu f c· p [n] − f c· p [n − 2] − ud [n − 2] − a1 d
− a2 d
(4.36)
∂a2
∂a2
∂a2
∂a2
(
)
We again define α 2 [n] = ∂ ∂a 2 u d · f [n] and calculate it recursively as
(
)
α 2 [n] = ∂ a 2 σu f c· p [n] − f c· p [n − 2] − ud · f [n − 2] − a1α 2 [n − 1] − a2α 2 [n − 2]
(4.37)
This value of α2[n] is therefore used in (4.29), instead of that given before in (4.26), for updating a2.
4.2.5. Partial derivative of εoei[n] with respect to φ0
The partial derivative of the error εoei[n] with respect to the transduction coefficient φ0 is given by
∂
∂
ε oei [n] =
− ic· p [n]
∂φ0
∂φ0
(
)
=
∂ φ0
ud [ n]
∂φ0 Reb
=
φ ∂
1
u d [ n] + 0
ud [ n]
Reb
Reb ∂φ0
(4.38)
Given the definition of ud[n] in (4.12), its derivative with respect to φ0 is given by
∂
∂
∂
u d [n] = σ u (ic·m [ n] − ic·m [n − 2]) − a1
u d [n − 1] − a 2
u d [ n − 2] .
∂φ 0
∂φ 0
∂φ 0
(4.39)
This is defined as ∂ φu [n] and calculated recursively as
∂ φu [n] = σu (ic·m [ n] − ic·m [n − 2]) − a1∂ φu [n − 1] − a2∂ φu [n − 2] .
(4.40)
Therefore, explicitly, the partial derivative of the error εoei[n] with respect to φ0 is
∂
φ
1
ε oei [n] =
ud [n] + 0 ∂ φu [n]
∂φ0
Reb
Reb
(4.41)
Loudspeaker system identification
117
where ∂ φu [n] is calculated recursively as per (4.40). Therefore the instantaneous estimate of the
gradient of the error surface along the parameter φ0 is
 1

φ
dˆφ0 [n] = ε[n] 
ud [n] + 0 ∂ φu [n]
Reb
 Reb

(4.42)
and thus the update equation
φ0 [n + 1] = φ0 [ n] − µ φ 0 dˆφ 0 [n]
(4.43)
is given explicitly by

 1
φ
u d [n] + 0 ∂ φu [n]
φ 0 [n + 1] = φ 0 [n] − µ φ0 ε[ n] 
Reb

 Reb
(4.44)
4.2.6. Convergence performance
The convergence performance of the electrical current output error adaptive algorithm has been
investigated for a variety of signals and initial values. These are presented in Figure 4.11 - Figure
4.28, as per Table 4.1.
Input signal
Initial an
val.
Time span
of plots
(page)
an plots
Noise, 0-2kHz
Upper right
30s
Noise, 0-2kHz
Upper left
30s
Noise, 0-2kHz
Lower left
30s
Noise, 0-2kHz
Lower right
46s
Speech, Male
Lower left
80s
Music
Lower left
160s
p. 118
p. 119
p. 120
p. 121
p. 122
p. 123
Figure 4.11
Figure 4.14
Figure 4.17
Figure 4.20
Figure 4.23
Figure 4.26
ζ, ω0, Reb, φ0
plots
Figure 4.12
Figure 4.15
Figure 4.18
Figure 4.21
Figure 4.24
Figure 4.27
Qtc, Qms, Err
plots
Figure 4.13
Figure 4.16
Figure 4.19
Figure 4.22
Figure 4.25
Figure 4.28
Table 4.1: Figure numbers for different input signals and settings of initial values of an.
The convergence performance using white noise is relatively good. The resonance frequency f0 is
identified (from an initial guess of at least 20% away from its actual value) in as quickly as two
seconds. Identification of the damping ratio ζ is slower, being identified in approximately 10 to 20
seconds, depending on whether the initial guess is above or below the actual value. The mechanical
and total resonance quality values, Qms and Qtc respectively, are determined from the damping ratio ζ,
and thus also require 10 to 20 seconds to be identified. One exception to these convergence times is
the case for which the initial guess of an is in the lower right corner of the tolerance quadrilateral,
shown in Figure 4.20 - Figure 4.22, on p. 121.
Convergence for the speech and music signals is considerably slower than for the white noise signal.
As per Figure 4.24, for the speech signal the resonance frequency is identified in about 10 seconds,
and the damping ratio is identified after 70 seconds. Approximately the same performance is seen for
the music signal. This may be a considerable problem for actual product implementation. Techniques
for obtaining faster convergence performance without increasing variance in the converged parameters
may be a subject of further research.
The accuracy of the values to which the parameters converged, by comparison to values measured
using other laboratory techniques, is quite good. Further detail is presented in §4.2.7.
Chapter 4
118
Convergence for initial values of a1, a2 in upper-right corner of tolerance quadrilateral
Solid: as updated
x: Final Value
Broken: Final value
−1.75
1
1
0.98
a
−1.8
0.96
0.94
5
10
15
20
25
30
a2
−1.85
0
0.92
1
0.9
0.98
a
2
0.88
0.96
0.94
0
0.86
5
10
15
20
Time (seconds)
25
30
0.84
−1.9
−1.8
a1
−1.7
Figure 4.11: [left] a1 and a2 vs. time; [right] a1 vs. a2.
R
zeta (Damping Ratio)
eb
15
Resistance (Ω)
Damping Ratio (%)
10
10
5
0
5
10
15
20
9
8
7
6
5
0
25
5
10
5
10
Resonance Frequency
15
φ0
20
25
15
20
Time (seconds)
25
Newton/Ampere
1200
1000
0
f (Hz)
1100
900
800
0
5
10
15
20
Time (seconds)
0.6
0.5
0.4
0.3
0
25
Figure 4.12: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Q )
tc
Qtc
10
5
5
10
15
Mechanical Q (Q
20
)
25
30
ms
150
Qms
100
50
0
0
Error (between measured and predicted current
10
15
0
0
(blocked)
ermspcnt
4
20
DC
3
10
2
10
1
10
0
5
10
15
20
Time (seconds)
25
30
10
0
5
10
15
20
Time (seconds)
25
30
Figure 4.13: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
Loudspeaker system identification
119
Convergence for initial values of a1, a2 in upper-left corner of tolerance quadrilateral
Solid: as updated
x: Final Value
Broken: Final value
−1.75
1
−1.8
a
1
0.98
−1.85
0.96
−1.9
0.94
5
10
15
20
25
30
a2
−1.95
0
0.92
1
0.9
a
2
0.88
0.95
0.86
0.9
0
5
10
15
20
Time (seconds)
25
30
0.84
−1.9
−1.8
a1
−1.7
Figure 4.14: [left] a1 and a2 vs. time; [right] a1 vs. a2.
Reb
zeta (Damping Ratio)
15
Resistance (Ω)
Damping Ratio (%)
10
10
5
0
5
10
15
20
9
8
7
6
5
0
25
5
10
5
10
Resonance Frequency
15
φ0
20
25
15
20
Time (seconds)
25
Newton/Ampere
1200
1000
0
f (Hz)
1100
900
800
0
5
10
15
20
Time (seconds)
0.6
0.5
0.4
0.3
0
25
Figure 4.15: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Q )
tc
tc
Q
4
5
10
15
Mechanical Q (Q
20
)
25
30
ms
150
Qms
100
50
0
0
Error (between measured and predicted current
10
6
2
0
(blocked)
ermspcnt
4
8
DC
3
10
2
10
1
10
0
5
10
15
20
Time (seconds)
25
30
10
0
5
10
15
20
Time (seconds)
25
30
Figure 4.16: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
Chapter 4
120
Convergence for initial values of a1, a2 in lower-left corner of tolerance quadrilateral
Solid: as updated
x: Final Value
Broken: Final value
1
−1.8
0.98
a
1
−1.78
0.96
−1.82
0.94
5
10
15
20
25
30
a2
−1.84
0
0.92
0.98
0.9
0.96
a
2
0.88
0.94
0.86
0.92
0.9
0
5
10
15
20
Time (seconds)
25
30
0.84
−1.9
−1.8
a1
−1.7
Figure 4.17: [left] a1 and a2 vs. time; [right] a1 vs. a2.
Reb
zeta (Damping Ratio)
15
Resistance (Ω)
Damping Ratio (%)
10
10
5
0
5
10
15
20
9
8
7
6
5
0
25
5
10
5
10
Resonance Frequency
15
φ0
20
25
15
20
Time (seconds)
25
Newton/Ampere
1200
1000
0
900
800
0
5
10
15
20
Time (seconds)
0.6
0.5
0.4
0.3
0
25
Figure 4.18: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Q )
tc
Qtc
2
5
10
15
Mechanical Q (Q
20
)
25
30
ms
15
Qms
10
5
0
0
(blocked)
10
4
0
0
DC
ermspcnt
3
6
Error (between measured and predicted current
f (Hz)
1100
2
10
1
10
0
5
10
15
20
Time (seconds)
25
30
10
0
5
10
15
20
Time (seconds)
25
30
Figure 4.19: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
Loudspeaker system identification
121
Convergence for initial values of a1, a2 in lower-right corner of tolerance quadrilateral
Note: time axis is from 0 to 46.5 secs.; previous graphs were from 0 to 30 secs.
Solid: as updated
x: Final Value
Broken: Final value
−1.6
1
−1.7
a
1
0.98
−1.8
0.96
−1.9
0.94
10
20
30
40
50
a2
−2
0
0.92
1
0.9
a
2
0.95
0.88
0.9
0.86
0.85
0.8
0
10
20
30
Time (seconds)
40
50
0.84
−1.9
−1.8
a
−1.7
1
Figure 4.20: [left] a1 and a2 vs. time; [right] a1 vs. a2.
R
zeta (Damping Ratio)
eb
15
Resistance (Ω)
Damping Ratio (%)
10
10
5
0
10
20
30
9
8
7
6
5
0
40
10
20
30
40
20
30
Time (seconds)
40
φ0
Resonance Frequency
Newton/Ampere
1200
1000
0
900
800
0
10
20
30
Time (seconds)
0.6
0.5
0.4
0.3
0
40
10
Figure 4.21: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Qtc)
tc
Q
2
10
20
30
Mechanical Q (Q )
40
20
30
Time (seconds)
40
50
ms
15
Qms
10
5
0
0
(blocked)
10
4
0
0
DC
ermspcnt
2
6
Error (between measured and predicted current
f (Hz)
1100
1
10
0
10
50
10
0
10
20
30
Time (seconds)
40
50
Figure 4.22: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
Chapter 4
122
Convergence for initial values of a1, a2 in lower-left corner of tolerance quadrilateral
Signal: Male speech; speech activity duty cycle: 50%.
Solid: as updated
x: Final Value
Broken: Final value
1
−1.8
0.98
a
1
−1.78
0.96
−1.82
0.94
10
20
30
40
50
60
70
80
a2
−1.84
0
0.92
0.98
0.9
0.96
a
2
0.88
0.94
0.86
0.92
0.9
0
10
20
30
40
50
Time (seconds)
60
70
0.84
80
−1.9
−1.8
a1
−1.7
Figure 4.23: [left] a1 and a2 vs. time; [right] a1 vs. a2.
Reb
zeta (Damping Ratio)
15
Resistance (Ω)
Damping Ratio (%)
10
10
5
0
10
20
30
40
50
60
9
8
7
6
5
0
70
10
20
30
50
60
70
10
20
30
40
50
Time (seconds)
60
70
Resonance Frequency
40
φ0
Newton/Ampere
1200
1000
0
900
800
0
10
20
30
40
50
Time (seconds)
60
0.6
0.5
0.4
0.3
0
70
Figure 4.24: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Q )
tc
Qtc
4
2
10
20
30
40
50
Mechanical Q (Q )
60
70
80
ms
20
Qms
15
10
5
0
0
(blocked)
10
6
0
0
DC
ermspcnt
2
8
Error (between measured and predicted current
f (Hz)
1100
1
10
0
10
−1
10
20
30
40
50
Time (seconds)
60
70
80
10
0
10
20
30
40
50
Time (seconds)
60
70
80
Figure 4.25: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
Loudspeaker system identification
123
Convergence for initial values of a1, a2 in lower-left corner of tolerance quadrilateral
Signal: Music1
Solid: as updated
x: Final Value
Broken: Final value
1
−1.7
a
1
0.98
−1.8
0.94
50
100
150
a2
−1.9
0
0.96
0.92
0.9
0.95
a
2
0.88
0.9
0.85
0
0.86
50
100
Time (seconds)
0.84
150
−1.9
−1.8
a
−1.7
1
Figure 4.26: [left] a1 and a2 vs. time; [right] a1 vs. a2.
R
zeta (Damping Ratio)
eb
15
Resistance (Ω)
Damping Ratio (%)
10
10
5
0
50
100
9
8
7
6
5
0
150
50
100
150
100
Time (seconds)
150
φ0
Resonance Frequency
Newton/Ampere
1200
1000
0
f (Hz)
1100
900
800
0
50
100
Time (seconds)
0.6
0.5
0.4
0.3
0
150
50
Figure 4.27: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Q )
DC
(blocked)
ermspcnt
tc
Q
tc
6
4
2
0
0
50
100
Mechanical Q (Q )
150
ms
Qms
15
10
5
0
0
50
100
Time (seconds)
150
Error (between measured and predicted current
2
10
1
10
0
10
0
50
100
Time (seconds)
150
Figure 4.28: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
1
The music used was a ‘popular music’ song, artist: DJ Spiller, title: Groovejet, © 2000 Positiva Records.
124
Chapter 4
4.2.7. Accuracy of the converged parameters for the current output-error form
The accuracy of the values to which adaptive algorithm converged has been assessed by comparison
between frequency response function (FRF) synthesised from the converged parameters, and actual
measurement of the FRF.
The voltage-to-displacement FRF was measured using a laser displacement vibrometer based on a
fringe-counter signal decoder. Further details of the experimental set-up are given in Appendix A.
This measured FRF is the red curve plotted in Figure 4.29 (magnitude) and in Figure 4.30 (phase).
The same voltage-to-displacement FRF has been synthesised using with the z-transform of (2.86)
(setting z = e iωTs ), using the values of Reb, φ0, a1, and a2 to which the adaptive algorithm converged.
This synthesised function is the blue curve plotted in Figure 4.29 (magnitude) and in Figure 4.30
(phase).
To ensure proper functioning of the prediction of the diaphragm-coil displacement xd[n], the predicted
displacement was fed to the analogue output of one channel of the processing hardware. This
‘predicted’ displacement signal was measured with the same frequency response analyser, and is
shown as the green curve in Figure 4.29 (magnitude) and in Figure 4.30 (phase). This predicted signal
(in green) closely corresponds to the synthesised FRF (in blue), and it is thus concluded that the
displacement is being predicted correctly.
As can be seen in Figure 4.29 and Figure 4.30, the measured value (in red) deviates from the value
synthesised from the converged parameters (in blue) at frequencies below 180Hz and at around
1.7kHz. The deviation below 180Hz is explained by a small leak in the rear cavity in which the
microspeaker was mounted. Below approximately 180Hz, the acoustic stiffness of the rear cavity is
lost, resulting in a higher receptance sensitivity. Above 180Hz, acoustic resistance of air flow in the
leak is high enough to render the leak negligible, and thus the effective acoustic stiffness of the cavity
appears in the measured receptance functions.
The aberration 1.7kHz is due to a rocking mode present in the microspeaker’s vibration dynamics.
This is due to the single-suspension construction of the microspeaker, as discussed in §2.1. This
phenomenon is discussed in detail in Appendix D.
The signal used for the FRF measurement was white noise, with a bandwidth of 0 to 2 kHz. As there is
no significant spectral content above 2.5kHz, the FRF measurements are very ‘noisy’ above this
frequency.
Loudspeaker system identification
125
Meas.:Red; FRF of real−time pred.:Green; Synth fr. adpt. Param.:Blue
−10
−15
dB re 1.0 mm / V
−20
−25
−30
−35
−40
2
3
10
10
Frequency Hz
Figure 4.29: Magnitude of measured receptance (red) vs. synthesised receptance from converged parameter
values (blue), and measurement of predicted displacement / voice-coil voltage, where the displacement is
predicted by the algorithm (green).
Meas.:Red; FRF of real−time pred.:Green; Synth fr. adpt. Param.:Blue
150
100
Phase (deg)
50
0
−50
−100
−150
2
3
10
10
Frequency (Hz)
Figure 4.30: Phase of measured receptance (red) vs. synthesised receptance from converged parameter values
(blue), and measurement of predicted displacement / voice-coil voltage, where the displacement is predicted
by the algorithm (green).
Chapter 4
126
4.2.8. Updating φk
In addition to the determination of parameters defining the linear properties of the loudspeaker, it may
also be interesting to see whether the nonlinear characteristics of the loudspeaker can be identified.
Here, the nonlinear property of the loudspeaker considered is the nonuniformity in the transduction
coefficient, as discussed in §2.2.1 part A.
Following the general strategy presented by Klippel (1999), the approach presented here is to define
nonuniformity of the transduction coefficient as a polynomial expansion in xd (the diaphragm-coil
displacement). The derivatives of the coefficients of this polynomial expansion are defined with
respect to an error signal. The polynomial coefficients are then iteratively updated using these
derivatives according to the standard LMS algorithm. The difference between the presentation here
and that given by Klippel is that an electrical current output-error signal as per Figure 4.10 is
considered here, instead of the velocity equation error signal used by Klippel.
An important initial problem to be considered is the derivative of φ(xd [n]) with respect to φk. Consider
the description of φ(x) in §2.2.1, part A, above wherein it is described as an Nφth-order polynomial
expansion in xd[n].
Given that xd[n] depends on φk, the derivative of φ(x d [n]) with respect to φk is given by
∂
∂
∂ 2
∂ Nφ
φ(xd [n]) = xdk [n] + φ1
xd [n] + φ2
xd [n] + ... + φ N φ
xd [n] .
∂φk
∂φk
∂φk
∂φk
(4.45)
The displacement xd[n] is calculated from the measured current ic·m[n], as described in (4.14) and
(4.13). This calculation involves the use of φ(x d [n]) . Therefore we treat xd[n] as a function of φk. In
this way, the derivative of the l th-power of xd[n] with respect to φk is
∂ l
∂
xd [n] = lxdl −1[n]
xd [n]
∂φk
∂φk
(4.46)
Given the formula for calculating the displacement xd[n] from the measured coil current ic·m[n] in
(4.14) and (4.13), the derivative of the displacement xd[n] with respect to φk is
∂
∂
∂
∂
xd [ n] = σ x
xd [n − 1] − a2
xd [n − 2] .
φ(xd [n − 1])ic·m [n] − a1
∂φk
∂φk
∂φk
∂φk
(4.47)
By defining this derivative as ∂ φ k xd [n] , it may be calculated recursively as so:
∂ φ k x d [ n] = σ x
∂
φ( xd [ n − 1])ic·m [n − 1] − a1∂ φ k x d [n − 1] − a2∂ φ k xd [n − 2] .
∂φk
(4.48)
With this definition the derivative of the l th-power of xd[n] with respect to φk in (4.46) may be
expressed as
∂ l
x d [ n] = l x dl −1 [n] ∂ φk xd [ n]
∂φ k
(4.49)
Substituting this expression into in the definition of the derivative of φ(xd [n]) with respect to φk in
(4.45) gives:
N
φ
∂
φ(xd [n]) = xdk [n] + ∂ φ k x d [n] φl l xdl −1[n]
∂φk
l =1
∑
(4.50)
Loudspeaker system identification
127
The summation term in (4.50) is in fact ∂ φ(xd [n]) ∂xd [ n] . An important feature of this summation is
that it is not specific to k. Therefore it need be calculated only once, and the same value may be used
for all of the φk derivatives. For this reason it is abbreviated as φ(∂x d [n]) such that (4.50) may be
expressed as
∂
φ(xd [n]) = xdk [n] + ∂ φ k xd [n]φ(∂xd [n])
∂φk
(4.51)
This definition makes possible the following explicit recursive formula for ∂ φ k xd [n] :
(
)
∂ φ k xd [n] = σ x xdk [n − 1] + ∂ φ k x d [n − 1]φ(∂xd [n − 1]) ic·m [n − 1] − a1∂ φ k x d [n − 1] − a2∂ φ k xd [n − 2] .
(4.52)
The importance of the ‘predictive’ nature of the discrete-time representation of the mechanical
mobility developed in §2.3.4 is revealed again in (4.52); were it not to be predictive, (4.52) would be
an implicit, and not an explicit equation.
For convenience, the derivative of φ(xd [n]) with respect to φk is abbreviated as ∂ φ k φ [n] .
With the above definitions, it is possible to consider the derivative the error εoei[n] with respect to φk.
The only term in the error which depends on φk is the velocity term ud[n], and thus:

∂
∂  φ(xm [n])


u
[
n
]
ε eoi [n] =
m
d
·

∂φk
∂φk  Reb

u [ n]
φ(xm [n])
= m·d ∂ φ k φ [ n] +
∂ φ k u [ n]
Reb
Reb
(4.53)
The term ∂ φk u [n] is the derivative of the velocity ud[n] with respect to φk. It is calculated recursively in
a manner similar to parametric derivatives of ud, e.g. an described in §4.2.2 , as so:
(
)
∂ φ k u [n] = σu ∂ φ k φ [n] ic·m [n] − ∂ φ k φ [n − 2] ic·m [ n − 2] − a1∂ φ k u [n − 1] − a2∂ φ k u [n − 2]
(4.54)
Summarising concisely, one first calculates for all k
φ(∂xd [n]) =
Nφ
∑ lφ x
l −1
l d [ n]
l =1
Then for each k, one calculates
∂ φ k xd [ n] = σ x ∂ φ k φ [n − 1] ic·m [n − 1] − a1∂ φ k xd [n − 1] − a2∂ φ k xd [ n − 2]
From which is calculated
∂ φ k φ [n] = xdk [n] + ∂ φ k xd [n]φ(∂xd [n])
With which is calculated
∂ φ k u [n] = σu ∂ φ k φ [n] ic·m [ n] − ∂ φ k φ [ n − 2] ic·m [ n − 2] − a1∂ φ k u [n − 1] − a2∂ φ k u [ n − 2]
(
)
With which one calculates
φ(xd [ n])
∂
u [ n]
ε eoi [n] = d ∂ φ k φ [n] +
∂ φ k u [ n]
∂φk
Reb
Reb
The convergence of values of the coefficients φk using this algorithm was found to be poor.
Convergence is either unstable or slow (depending on the convergence parameter values), and did not
128
Chapter 4
consistently converge to the same values for the same loudspeaker & signal. At the time of writing it
was not clear why good convergence could not be achieved. Further investigation of techniques for
determining values of φk using this method is suggested as a topic for further research.
Loudspeaker system identification
129
4.3. Voltage output-error form
We consider a traditional output error form where an error signal is derived form
εoev [n] = vc·m [n] − vc· p [n]
(4.55)
where vc·m[n] is the measured voltage signal, and vc·p[n] is the predicted voltage signal. A block
diagram of this form for system identification is shown in Figure 4.31.
adt·1 bdt·1
z
-1
Σ
Σ
bdt·0
φ(xd)
ic·m[n]
Leb
Reb
vc·m [n]
φ(xd)
+
Σ
fc·p[n]
-
Linear mechanical
model
ud[n]
xd[n]
Σ
vc·p[n]
-
+
Σ
εoev[n]
k1(xd)
Figure 4.31: Block diagram of the voltage output error form for loudspeaker system identification. Notice that
this form of the updating algorithm includes Leb, as per (4.57), below.
The ‘measured’ voltage signal vc·m[n] may be measured, i.e. sampled with an A/D converter.
Alternatively, if the power amplifier is assumed to be linear and its gain is known, it may be calculated
from the input signal.
The predicted voltage signal is calculated from
v c· p [n] = R eb i c·m [ n] + φ(x d [n]) u d [ n]
(4.56)
where ic·m[n] is the measured current signal, and ud[n] is the velocity predicted from the force equation.
The velocity signal ud[n] is iteratively calculated in the same manner as for the current output error,
defined in Eq. (4.12), above. As a result, the derivatives of φ0, φk, and a1 and a2 with respect to the
error for this voltage output-error form are similar as they were for the current output-error form.
It is possible to include the effects of a blocked electrical inductance quite easily. By making an
approximate differentiation of the electrical current, the effects of the electrical inductance may be
included as follows:
v c· p [n] = Reb i c·m [n] + Leb hdt [n] ∗ i c·m [n] + φ(x d [ n]) u d [ n]
(4.57)
where hdt[n] is the impulse response of a differentiation approximation, as in (2.84) presented in
§2.3.4.
Chapter 4
130
4.3.1. Parameter updating
Updating Reb
∂
εoev [n] = −ic·m [ n]
∂Reb
(4.58)
The Reb estimate is therefore updated by
Reb [n + 1] = Reb [n] + µ Reb ε[n] ic·m [n]
(4.59)
Updating Leb
The derivative of the error is with respect to Leb is given by
∂
εoev [n] = −hdt [n] ∗ ic·m [n]
∂Leb
(4.60)
The Leb estimate is therefore updated by
Leb [n + 1] = Leb [n] + µ Leb ε eov [n]hdt [n] ∗ ic·m [n]
(4.61)
Updating ak
As noted above, neither the measured current ic·m[n] nor the predicted force fc·p[n] depend on the
feedback coefficients ak. Therefore the derivative of the error with respect to ak is that of a standard
output-error IIR algorithm where the IIR filter concerned is that for the mechanical mobility. Thus
generally one has
∂
∂
ε eov [n] =
− v c · p [ n]
∂a k
∂a k
(
)
∂
= −φ 0
u d · f [ n]
∂a k
(4.62)
The derivative of the ud·f[n] with respect to the feedback coefficients ak here is the same as for the
electrical current output error form defined in §4.2. Thus the instantaneous estimate of the gradient of
the error surface along the parameter ak is therefore given by
∂ε [n]
= ε[n]φ0α k [n]
dˆa k [n] = −ε[n] eov
∂ak
(4.63)
where αk[n] is calculated iteratively as per (4.26). The ak parameters are updated according to the LMS
algorithm as so:
ak [n + 1] = ak [ n] − µ a k dˆa k [n]
(4.64)
where µ ak is a convergence parameter specific to ak. Summing up explicitly, this is
ak [n + 1] = ak [n] − µ a k ε[n]φ0α k [ n]
Updating φ0
The partial derivative of the error εoev[n] with respect to the transduction coefficient φ0 is given by
(4.65)
Loudspeaker system identification
131
∂
∂
ε oev [n] =
− v c· p [ n]
∂φ 0
∂φ 0
(
=−
)
∂
φ 0 u d · f [ n]
∂φ 0
= −u d · f [n] − φ 0
(4.66)
∂
u d · f [ n]
∂φ 0
The derivative of ud·f[n] with respect to φ0 is defined as ∂ φu [n] , and is calculated iteratively as per
(4.40). Therefore, explicitly, the partial derivative of the error εoev[n] with respect to φ0 is
∂
εoev [n] = −ud · f [n] − φ0∂ φu [n]
∂φ0
(4.67)
Therefore the instantaneous estimate of the gradient of the error surface along the parameter φ0 is
(
dˆφ 0 [n] = ε[n] − u d · f [n] − φ 0∂ φu [ n]
)
(4.68)
and thus the update equation
φ0 [n + 1] = φ0 [ n] − µ φ 0 dˆφ 0 [n]
is given explicitly by
(
(4.69)
)
φ0 [n + 1] = φ0 [n] − µφ 0 ε[n] ud · f [n] + φ0∂ φu [n]
(4.70)
Updating φk
The derivative of the kth coefficient of φ(x) for the voltage output error form is
∂
∂
(− φ(x d [n])u d [n])
ε eov [n] =
∂φ k
∂φ k
(4.71)
= −u d [n]∂ φ k φ [n] − φ( x d [n])∂ φ k u [n]
where ∂ φk φ [n] and ∂ φk u [n] are as defined for the electrical current output error form in §4.2.8.
4.3.2. Convergence performance
The convergence performance of the electrical current output error adaptive algorithm has been
investigated for a variety of signals and initial values. These are presented in Figure 4.32 -Figure 4.46,
as per Table 4.2.
Input signal
Noise, 0-2kHz
Noise, 0-2kHz
Noise, 0-2kHz
Speech, Male
Music
Initial an
Leb
Duration (page)
adaptation
val.
Upper left
No
30s
p. 133
Target
Yes
30s
p. 134
Lower left
Yes
30s
p. 135
Lower left
Yes
142s p. 136
Lower left
Yes
145s p. 137
an plots
Figure 4.32
Figure 4.35
Figure 4.38
Figure 4.41
Figure 4.44
ζ, ω0, Reb, φ0
plots
Figure 4.33
Figure 4.36
Figure 4.39
Figure 4.42:
Figure 4.45
Qtc, Qms, Err
plots
Figure 4.34
Leb plot
-
-
Figure 4.37
Figure 4.40 Figure 4.38
Figure 4.43 Figure 4.41
Figure 4.46 Figure 4.44
Table 4.2: Figure numbers for different input signals and settings of initial values of an.
The convergence of the voltage output error algorithm is not appreciably different from the electrical
current output error form. The resonance frequency tends to converge quickly, and whereas the
132
Chapter 4
damping factor (and the Q-values derived from it) take longer to converge, up to 20 seconds with
white noise, and up to 60 seconds with speech and music signals.
The additional parameter identified by this form of the algorithm, the blocked electrical inductance
Leb, converges quite quickly, as shown in Figure 4.37 (left side). Furthermore, the convergence of Leb
does not have a significant effect on the value of other converged parameters. This can be seen from
the figures on page 134. At t = 5.64 seconds, the convergence of the Leb is begun. Other parameters
had been allowed to converge to their final values, before convergence of Leb was begun. As can be
seen in the figures on page 134, the convergence of Leb does not significantly affect the converged
values of the other parameters. This is important, as it indicates that the non-convergence of Leb does
not cause bias errors in the values to which other parameters converge.
The parameter spread in Leb for the speech and music signals is very large, up to 50% for the music
signal. This is considered unsatisfactory performance. Lower spread in the converged value of Leb
would be needed if speech or music signals are used during parameter determination. This is
suggested as a possible subject for further research.
Loudspeaker system identification
Solid: as updated
133
Broken: Final value
x: Final Value
1
−1.7
0.98
a
1
−1.75
−1.8
0.96
−1.85
0.94
5
10
15
20
25
a2
−1.9
0
0.92
0.9
0.98
0.96
0.88
a
2
0.94
0.92
0.86
0.9
0.88
0
5
10
15
20
Time (seconds)
25
0.84
−1.9
−1.8
a
−1.7
1
Figure 4.32: [left] a1 and a2 vs. time; [right] a1 vs. a2.
R
eb
10
0
Resistance (Ω)
Damping Ratio (%)
zeta (Damping Ratio)
14
12
10
8
6
4
2
5
10
15
20
9
8
7
6
5
0
25
5
10
5
10
Resonance Frequency
15
φ0
20
25
15
20
Time (seconds)
25
Newton/Ampere
1200
1000
0
f (Hz)
1100
900
800
0
5
10
15
20
Time (seconds)
0.6
0.5
0.4
0.3
0
25
Figure 4.33: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Qtc)
DC
(blocked)
Percentage Error
2
10
6
Q
tc
4
0
0
5
10
15
Mechanical Q (Q
20
)
25
15
20
Time (seconds)
25
ms
Error (%)
2
1
10
Qms
100
50
0
0
0
5
10
10
0
5
10
15
20
Time (seconds)
25
Figure 4.34: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
Chapter 4
134
Effect of including adaptation of Leb on other parameters. At t= 5.64 seconds, the convergence parameter for
Leb is set from zero to its regular value. There is a slight effect on the determined values of ζ and φ0. [Note
that Figure 4.37 (bottom left) shows the adapted value of Leb and not Qtc and Qms as in previous figures.]
Solid: as updated
Broken: Final value
x: Final Value
1
−1.7
0.98
a
1
−1.75
−1.8
0.96
−1.85
0.94
5
10
15
20
25
a2
−1.9
0
0.92
0.9
0.98
0.96
0.88
a
2
0.94
0.92
0.86
0.9
0.88
0
5
10
15
20
Time (seconds)
0.84
25
−1.9
−1.8
a
−1.7
1
Figure 4.35: [left] a1 and a2 vs. time; [right] a1 vs. a2.
Reb
10
0
Resistance (Ω)
Damping Ratio (%)
zeta (Damping Ratio)
14
12
10
8
6
4
2
5
10
15
20
9
8
7
6
5
0
25
5
10
5
10
Resonance Frequency
15
φ0
20
25
15
20
Time (seconds)
25
Newton/Ampere
1200
1000
0
900
800
0
5
10
15
20
Time (seconds)
0.6
0.5
0.4
0.3
0
25
Figure 4.36: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
L
eb
Percentage Error
2
10
60
Inductance (µH)
50
40
Error (%)
f (Hz)
1100
30
1
10
20
10
0
0
0
5
10
15
time (seconds)
20
25
Figure 4.37: [left]: Leb vs. time [right]: Percentage error.
10
0
5
10
15
20
Time (seconds)
25
DC
(blocked)
Loudspeaker system identification
135
Convergence for initial values of a1, a2 in lower-left corner of tolerance quadrilateral.
Including adaptation of Leb.
Solid: as updated
L
Broken: Final value
eb
60
−1.7
−1.75
a
1
50
−1.8
−1.9
0
5
10
15
20
25
0.98
Inductance (µH)
−1.85
40
30
20
0.96
a
2
0.94
10
0.92
0.9
0.88
0
5
10
15
20
Time (seconds)
0
0
25
5
10
15
time (seconds)
20
25
20
25
15
20
Time (seconds)
25
Figure 4.38: [left] a1 and a2 vs. time; [right] Leb vs. time.
R
eb
10
0
Resistance (Ω)
Damping Ratio (%)
zeta (Damping Ratio)
14
12
10
8
6
4
2
5
10
15
20
9
8
7
6
5
0
25
5
10
5
10
Resonance Frequency
15
φ0
Newton/Ampere
1200
1000
0
900
800
0
5
10
15
20
Time (seconds)
0.6
0.5
0.4
0.3
0
25
Figure 4.39: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Q )
tc
DC
(blocked)
Percentage Error
2
10
Q
tc
6
4
2
0
0
5
10
15
Mechanical Q (Q
20
)
25
15
20
Time (seconds)
25
ms
Error (%)
f (Hz)
1100
1
10
20
Qms
15
10
5
0
0
0
5
10
10
0
5
10
15
20
Time (seconds)
25
Figure 4.40: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
Chapter 4
136
Convergence for initial values of a1, a2 in lower-left corner of tolerance quadrilateral
Signal: Male speech; speech activity duty cycle: 50%.
Solid: as updated
Leb
Broken: Final value
40
−1.7
−1.75
a
1
35
−1.8
30
−1.9
0
20
40
60
80
100
120
140
0.98
Inductance (µH)
−1.85
25
20
15
0.96
10
a
2
0.94
0.92
5
0.9
0.88
0
20
40
60
80
Time (seconds)
100
120
0
0
140
20
40
60
80
time (seconds)
100
120
140
100
120
140
100
120
140
Figure 4.41: : [left] a1 and a2 vs. time; [right] Leb vs. time.
Reb
10
0
Resistance (Ω)
Damping Ratio (%)
zeta (Damping Ratio)
14
12
10
8
6
4
2
20
40
60
80
100
120
9
8
7
6
5
0
140
20
40
60
80
φ0
Resonance Frequency
Newton/Ampere
1200
1000
0
900
800
0
20
40
60
80
Time (seconds)
100
120
0.6
0.5
0.4
0.3
0
140
20
40
60
80
Time (seconds)
Figure 4.42: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Qtc)
DC
(blocked)
Percentage Error
2
10
Qtc
6
4
2
0
0
20
40
60
80
Mechanical Q (Q
100
120
140
)
ms
Error (%)
f (Hz)
1100
1
10
20
Qms
15
10
5
0
0
0
20
40
60
80
Time (seconds)
100
120
140
10
0
20
40
60
80
Time (seconds)
100
120
140
Figure 4.43: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
Loudspeaker system identification
137
Convergence for initial values of a1, a2 in lower-left corner of tolerance quadrilateral
Signal: Music1
Solid: as updated
L
Broken: Final value
eb
40
−1.7
−1.75
a
1
35
−1.8
30
−1.9
0
20
40
60
80
100
120
140
0.98
Inductance (µH)
−1.85
25
20
15
0.96
10
a
2
0.94
0.92
5
0.9
0.88
0
20
40
60
80
100
Time (seconds)
120
140
0
20
40
60
80
100
time (seconds)
120
140
100
120
140
60
80
100
Time (seconds)
120
140
Figure 4.44: [left] a1 and a2 vs. time; [right] Leb vs. time.
Reb
10
0
Resistance (Ω)
Damping Ratio (%)
zeta (Damping Ratio)
14
12
10
8
6
4
2
20
40
60
80
100
120
9
8
7
6
5
0
140
20
40
60
80
φ
Resonance Frequency
0
Newton/Ampere
1200
1000
0
f (Hz)
1100
900
800
0
20
40
60
80
100
Time (seconds)
120
0.6
0.5
0.4
0.3
0
140
20
40
Figure 4.45: [left, upper]: Damping ratio (ζ); [left, lower]: Resonance Frequency; [right, upper]
electrical resistance (Reb); [right, lower]: Transduction coefficient (φ0).
Total Q (Q )
tc
DC
(blocked)
Percentage Error
2
10
Q
tc
6
4
0
0
20
40
60
80
Mechanical Q (Q
100
)
120
60
80
100
Time (seconds)
120
140
ms
Error (%)
2
1
10
20
Qms
15
10
5
0
0
0
20
40
140
10
0
20
40
60
80
100
Time (seconds)
120
140
Figure 4.46: [left, upper]: Total Q-factor (Qtc); [left, lower]: Mechanical Q-factor; [right]: Percentage error.
1
The music used was a ‘popular music’ song, artist: DJ Spiller, title: Groovejet, © 2000 Positiva Records.
Chapter 4
138
4.4. Displacement equation-error form
The displacement equation error form is considered as an alternative to the velocity equation error
form presented by Klippel. The displacement must be calculated for nonlinear parameter updating in
Klippel’s form (1999). Thus it would be simpler to use the displacement signal as the basis of an error
function outright.
4.4.1. Displacement from the voltage equation
The displacement is predicted by the voltage equation by isolating and integrating the velocity signal.
The velocity signal may be predicted by the voltage equation according to:
u d ·v [ n] =
1
(vc [n] − Rebic [n])
φ( x)
(4.72)
This signal is integrated to provide a displacement signal.
xd ·v [n] = h dt [n] ∗ ud ·v [n]
∫
(4.73)
The convolution in (4.73) will be implemented as a difference equation. Specific details of this
difference equation directly impact convergence performance of the algorithm, and are thus addressed
below.
4.4.2. Displacement from the force equation
The displacement may be calculated from the force according to the mechanical receptance defined in
§2.3.2 as per:
xd · f [n] = f c· p [n] ∗ hX m [n]
= σ x fc· p [n − 1] − a1xd · f [n − 1] − a2 xd · f [n − 2]
(4.74)
The force fc·p[n] may be calculated from the measured electrical current as so
(
)
(
)
f c· p [n] = φ xd · f [n] ic·m [n] − k1 xd · f [n] xd · f [n]
(4.75)
Again, the importance of the predictive nature of the displacement difference equation (4.74) appears.
The present sample of the displacement xd·f[n] is calculated from the previous samples of the ‘input’
(the force) fc·p[n]. If (4.74) were not ‘predictive,’ it would form with (4.75) an implicit, and not
explicit, equation pair.
4.4.3. Error definition
The error is calculated by the difference between the displacement calculated from the voltage
equation and the displacement calculated from the force equation.
εeed [n] = xd ·v [n] − xd · f [n]
= h dt [n] ∗ ud ·v [n] − σ x f c· p [ n − 1] − a1 xd · f [n − 1] − a2 xd · f [n − 2]
∫
(4.76)
For the linear case, (4.76) reduces to
ε eed [n] = h dt [n] ∗ ud ·v [n − 1] − σ x φ0ic [n − 1] + a1 xd · f [n − 1] + a2 xd · f [n − 2]
∫
(4.77)
Substituting velocity ud·v[n] with the RHS of (4.72) provides the following explicit difference equation
for the error:

 1
ε eed [n] = h dt [n] ∗  (vc·m [n − 1] − Rebic·m [n − 1]) − σ x φ0ic [n − 1] + a1 xd · f [n − 1] + a2 xd · f [n − 2]
∫

 φ0
(4.78)
Loudspeaker system identification
139
4.4.4. Parameter updating – linear case
Updating Reb
•
•
ic·m[n] and vc·m[n] are input signals, and thus do not depend on Reb
xd·f [n] is calculated from the force equation and thus does not depend on Reb
For the purpose of integration, a discrete-time integrator impulse response in the form of an IIR filter,
using a z-domain transfer function determined according to the bilinear transform:
 1 − z −1 
h dt [n] = Z −1  21F
−1 
s
∫
 1+ z 
(4.79)
In this case, (4.73) becomes this difference equation:
xd ·v [n] = bint 0ud ·v [ n] + bint 1ud ·v [n − 1] − aint 1 xd ·v [n]
(4.80)
The derivative of the error with respect to Reb is therefore
∂
∂
∂
∂
εeed [n] = bint·0
ud ·v [n] + bint·1
ud ·v [ n − 1] − aint 1
xd ·v [ n − 1]
∂Reb
∂Reb
∂Reb
∂Reb
(4.81)
With ud·v[n] defined, for the linear case the derivative of the error with respect to Reb will be
∂
b
b
∂
εeed [n] = − int·0 ic·m [n] − int·1 ic·m [n − 1] − aint 1
xd ·v [n − 1]
∂Reb
φ0
φ0
∂Reb
(4.82)
Noting that ∂x d ·v [n] ∂Reb is the same as ∂ε eed [n] ∂Reb , the partial derivative of the error with respect
to Reb is calculated recursively by:
ε∂Reb [n] =&
=−
∂
εeed [n]
∂Reb
bint·0
b
ic·m [n] − int·1 ic·m [n − 1] − aint·1ε∂Reb [n − 1]
φ0
φ0
(4.83)
From this the instantaneous estimate of the gradient of the error surface along Reb is given by
dˆReb [n] = εeed [n]ε∂Reb [n]
(4.84)
Thus the LMS algorithm provides the following parameter-update equation
Reb [n + 1] = Reb [n] − µ Reb εeed [n]ε∂Reb [n]
(4.85)
It is found that this update method does not lead to proper convergence. Misalignment of Reb
accumulates in the estimate of xd·v[n], because its estimation is performed by integration of an equation
containing Reb, i.e. Eqs. (4.72) - (4.80). This problem is illustrated by simulation, wherein the input
voltage is white noise. The results of this simulation are shown in Figure 4.47.
Chapter 4
140
x : blue
dv
x : Green
df
600
0.4
Error
0.2
0
−0.2
200
−0.4
0
5
10
15
5
10
Time (seconds)
15
0
200
−200
Reb (Ohms)
Predicted Displacement (mm)
400
−400
0
−200
−600
0
5
10
Time (seconds)
15
20
0
Figure 4.47: LEFT: Displacement predicted from force equation (Green) and from voltage equation (blue);
UPPER RIGHT: Error; LOWER RIGHT: Updated Reb (blue) and actual value (8.0, in green).
A filtered-error version of the LMS algorithm was investigated as a potential solution to this problem.
By low-pass filtering the error used to update Reb, this problem can be eliminated. Convergence
results using this filtered-error approach are shown in Figure 4.48. As can be seen in this figure, this
results in convergence of Reb to its correct value. However, error in the displacement predicted from
the voltage xd·v[n] equation accumulates, and does not fade-away after Reb has converged to its proper
value. A different approach is still needed.
x : blue
dv
x : Green
−4
df
x 10
Filtered Error
10
6
5
0
−5
0
5
10
15
5
10
Time (seconds)
15
4
8
2
(Ohms)
Predicted Displacement (mm)
8
R
eb
0
−2
0
5
10
Time (seconds)
15
20
7
6
5
4
0
Figure 4.48: Filtered-error approach to adaptive Reb; LEFT: Displacement predicted from force equation
(Green) and from voltage equation (blue); UPPER RIGHT: Error; LOWER RIGHT: Updated Reb (blue) and actual value
(8.0, in green).
In order to also eliminate the accumulation of error in xd·v[n], a limited-memory integration
approximation is used when predicting it from ud·v[n]. This can be done by creating a modified
integration filter whose z-transform is given by the product between the integrator and a first-order
high-pass filter, as so:
 1 + z −1

h dt ·lm [n] = Z −1  21F
H HP ( z )
−
1
s
∫
 1− z

(4.86)
Loudspeaker system identification
141
In this way, the displacement determined by the voltage equation becomes
xd ·v [n] = bint·0ud ·v [n] + bint·1ud ·v [n − 1] + bint·2ud ·v [n − 2] − aint 1 xd ·v [n − 1] − aint 2 xd ·v [n − 2]
(4.87)
where the coefficients (bint·0 … aint·2) are determined from the inverse z-transform in (4.86). Thus the
recursive calculation of the partial derivative of the error with respect to Reb becomes:
ε∂Reb [n] =&
∂
εeed [ n]
∂Reb
b
b
b
= − int·0 ic [n] − int·1 ic [n − 1] − int·2 ic [n − 2] − aint·1ε∂Reb [n − 1] − aint·2ε∂Reb [n − 2]
φ0
φ0
φ0
(4.88)
The LMS algorithm uses this to update the estimate of Reb in the same manner as (4.85).
The convergence performance of Reb using this limited-memory integrator is shown in Figure 4.49. As
can be seen, the displacement predicted from the voltage equation does have the accumulated error as
it did before. Also, the convergence of Reb is much faster and smoother.
xdv: blue
xdf: Green
−4
x 10
0.5
4
Error
2
0.3
0
−2
0.2
0.1
0
5
10
15
5
10
Time (seconds)
15
0
8
−0.1
Reb (Ohms)
Predicted Displacement (mm)
0.4
7
−0.2
6
−0.3
−0.4
0
5
5
10
Time (seconds)
15
20
4
0
Figure 4.49: Limited-memory integration of ud·v[n]; LEFT: Displacement predicted from force equation (Green)
and from voltage equation (blue); UPPER RIGHT: Error; LOWER RIGHT: Updated Reb (blue) and actual value (8.0, in
green).
Updating a1, a2 – linear case
∂xd · f
∂
∂εeed [n]
− σ x φ0ic·m [n] + a1 xd · f [n − 1] + a2 xd · f [n − 2]
=−
=
∂ak
∂ak
∂ak
[
]
(4.89)
Given that ic·m[n] is a measured signal, its derivative with respect to ak is zero. From the definition of
σx in (2.67), its derivative with respect to a1 is given by
∂σ x 2 + a2
=
∂a1
kt
(4.90)
and similarly for a2. As discussed in §2.4, kt cannot be known a-priori. It is thus written as
∂σ x
2+a
= 2 22
∂a1 ωz Fs mt
where ωz is determined from a1 and a2 as per (2.73). Therefore (4.89) becomes, for a1 and a2:
(4.91)
Chapter 4
142
−
−
∂xd · f
∂a1
∂xd · f
∂a2
=−
2 + a2
∂
∂
+ xd · f [n − 1] + a1
xd · f [ n − 1] + a2
xd · f [n − 2]
2 2
∂a1
∂a1
ωz Fs mt
(4.92)
=−
2 + a1
∂
∂
+ xd · f [n − 2] + a1
xd · f [ n − 1] + a2
xd · f [n − 2]
2 2
∂a2
∂a2
ωz Fs mt
(4.93)
Using the same principle as the LMS IIR algorithm, the derivative of the error with respect to ak is
defined as αk ,
∂ε eed [ n]
∂a k
α k [ n] =
=−
∂x d · f [n]
(4.94)
∂a k
With this definition, this derivative may be recursively calculated according to (4.92) as so:
α1[n] =
2 + a2
− xd · f [n − 1] − a1α1[ n − 1] − a2α1[n − 2]
ω2z Fs2 mt
(4.95)
and similarly for α2[n]. The instantaneous estimate of the gradient of the error surface along the
parameter ak is therefore given by
∂ε [n]
dˆak [n] = εeed [n] eoi
∂ak
(4.96)
= εeed [n]α k [n]
where αk[n] is calculated iteratively as per (4.95). The ak parameters are updated according to the LMS
algorithm as so:
ak [n + 1] = ak [ n] − µ a k dˆa k [n]
(4.97)
where µ ak is a convergence parameter specific to ak. Summing up explicitly, this is
ak [n + 1] = ak [ n] − µ a k εeed [n]α k [ n]
(4.98)
Updating φ0 – linear case
The derivative of the error with respect to φ0 is complicated by its presence in the voltage equation.
Due to the method of integration needed for proper convergence of Reb described in (4.87), the term φ0
appears ‘buried’ in this difference equation. Thus the derivative of φ0 with respect to xd·v[n[ is
considered separately from xd·f[n], which are defined as so:
∂xd ·v [n]
=& ∂ φ·v [n]
∂φ0
∂xd · f [n]
∂φ0
=& ∂ φ· f [n]
(4.99)
Loudspeaker system identification
143
The derivative of the displacement as determined from the voltage equation, ∂ φ·v [n] , is considered
first. With xd·v[n] as defined in (4.87), the derivative is given by
∂ φ·v [n] = bint·0
∂ud ·v [n]
∂u [n − 1]
∂u [n − 2]
+ bint·1 d ·v
+ bint·2 d ·v
− aint·1∂ φ·v [n − 1] − aint·2∂ φ·v [n − 2] (4.100)
∂φ0
∂φ0
∂φ0
With ud·v[n] as given in (4.72), its derivative with respect to φ0 is given by
∂ud ·v [n]
1
= − 2 (vc·m [n] − Rebic·m [n])
φ0
φ0
(4.101)
Note that the RHS of this is the same as − u d ·v [ n] φ 0 . The derivative of xd·v[n] with respect to φ0 is
therefore given by the following recursive calculation
∂ φ ·v [ n ] = −
1
(bint·0ud ·v [n] + bint·1ud ·v [n − 1] + bint·2ud ·v [n − 2]) − aint·1∂ φ·v [n − 1] − aint·2∂ φ·v [n − 2] (4.102)
φ0
The derivative of φ0 with respect to the displacement determined from the force equation is recursively
determined in a manner similar to the ak coefficients, as so:
∂ φ· f [n] = σ xic·m [n] − a1∂ φ· f [n − 1] − a2∂ φ· f [ n − 2]
(4.103)
The derivative of the error is therefore given by
∂εeed [ n]
= ∂ φ ·v [ n ] − ∂ φ · f [ n ]
∂φ0
resulting in the update equation
(4.104)
(
)
φ0 [n + 1] = φ0 [ n] − µ φ 0 εeed [n] ∂ φ·v [n] − ∂ φ· f [ n]
(4.105)
It is noted that the calculation of the derivative of the error with respect to φ0 is considerably more
complicated for this displacement equation-error form than it is for either the current or the voltage
output error form described above. As this complication will become manifold worse when the
nonlinear case is concerned, the displacement equation error form has not been pursued further.
4.5. Conclusions regarding system identification
Three methods are investigated for determination of linear parameters of a loudspeaker. The methods
are developed specifically to determine those loudspeaker parameters which are subject to drift, and
thus cannot be known a priori, as discussed in §2.4. This is made possible because one of the basic
parameters of the loudspeaker can the total moving mass be known a priori (because it is not subject
to drift). This permits system identification of the loudspeaker using a method that only considers it’s
electrical characteristics, without the need for a vibration measurement.
The displacement equation error form investigated in §4.4 is considered too complicated, and is thus
not investigated in full detail. The electrical current and voltage output error forms, presented in §4.2
and §4.3 respectively, were both found to be sufficiently simple. Full investigation of these two
algorithms is made, including measurements of their convergence performance using signals obtained
from an actual loudspeaker. Convergence performance of both algorithms was found to be good for
white noise signals. Convergence performance with speech and music signals was found to be
similarly accurate, but considerably slower. This slow convergence rate for the speech and music
signals may prove problematic in some applications.
144
Chapter 4
It is possible that other common adaptive algorithms, such as the filtered-X or filtered-error LMS, or
the recursive-least-squares algorithm, may increase the convergence rate with speech and music
signals. This will of course come at the cost of an increase in the algorithm’s complexity. This may
be a suitable subject for further research.
Little difference in the convergence performance is found between the electrical current and voltage
output error forms. The voltage output error form is slightly simpler than the electrical current output
error form, and thus seems to have the advantage between the two.
Attempts were made to identify coefficients of a polynomial approximation to the nonuniformity of
the transduction coefficient. This is tried with both the electrical current and voltage output error
forms. Convergence of the coefficient parameters was found to be erratic. Further research on this
subject will be needed if determination of these parameters is deemed necessary.
Parameter determination algorithms have been evaluated only for the case of a loudspeaker mounted
in a closed box. More complex acoustic enclosures, such as a vented enclosure, would require a more
complicated model of the linear dynamics. For example, in the case of a vented enclosure, an
additional second-order IIR section would be needed to model the extra pole pair in the linear
dynamics of the vented box system. This may also be a suitable subject for further research.
4.6. References
Johnson, C. R. Jr., and M. G. Larimore, “Comments on and additions to ‘An adaptive recursive LMS
filter,” Proceedings of the IEEE, 65, pp. 1399-1402. (Sept. 1977)
Klippel, Wolfgang J., “Nonlinear Adaptive Controller for Loudspeakers with a Current Sensor,”
presented at the 106th Convention of the AES (May 8-11, 1999), preprint no. 4864; J. Audio Eng.
Soc. (Abstracts), 47, p. 512 (Jun. 1999)
Knudsen, M. H., J. Grue Jensen, V. Julskjær, and Per Rubak, “Determination of Loudspeaker Driver
Parameters Using a System Identification Technique,” Journal of the Audio Eng. Soc., 37, pp. 700708. (Sept. 1989)
Ljung, Lennart, System Identification: Theory for the User, Prentice Hall Ptr., Upper Saddle River, NJ,
USA. (1999)
Oppenheim, Alan V., and Ronald W. Schafer, Discrete-time Signal Processing, Prentice Hall,
Englewood Cliffs, New Jersey, USA. (1989)
145
5.
Applications of active control of a loudspeaker
Applications of active control for a loudspeaker are presented in this chapter. In §5.1, applications of
linear control are presented. In §5.2, applications of nonlinear control are presented.
The presentation of these applications assumes the active control system is implemented as an
adaptive feedforward controller, as shown in Figure 1.4, and in more detail in Figure 4.1. As per these
figures, the feedforward processor is somewhat separate from the system identification processing
algorithms. The feedforward processor in this context may be discussed in the same manner as in the
non-adaptive feedforward control system, with one important difference. In the adaptive feedforward
controller, algorithms for feedforward processing must be designed so that they can be easily updated
from the results of system identification algorithms. Particular attention has been given to this
requirement in the development and presentation feedforward processing algorithms in this chapter.
The algorithms have been designed such that they may updated with parameters from the system
identification algorithms, without the need for any type of transformation on these parameters.
The linear control applications presented in §5.1 are limited to low-order equalisation. This is
straightforward equalisation, for extending the low-frequency response and controlling the total Qvalue of a loudspeaker mounted in a closed box. The focus of discussion in §5.1 is on the digital
implementation of the equaliser, and how this equaliser can be updated using the parameters of the
loudspeaker identified by the system identification algorithms presented in Chapter 4.
In §5.2, the use of nonlinear control is considered. Specifically, it is considered how nonlinear control
can be used to improve the overall sensitivity of a loudspeaker. The compromise between reduction of
coil height and additional amplifier output required for nonlinear distortion compensation is
investigated. Results from a series of simulations are presented, showing the net sensitivity of an
electroacoustic system for different coil heights, at different vibration displacements, including the
additional output required for nonlinear compensation. To test the suggestions from these simulations,
loudspeakers were prepared with shortened voice coil heights. The simple algorithm for compensation
of nonlinear distortion, developed in §3.3.7, is used to compensate for nonlinear distortion in these
shortened-voice-coil-height loudspeakers. Measurements of the linear frequency response of these
shortened-voice-coil-height loudspeakers are shown, indicating the small-signal sensitivity increase
provided by shortening the voice-coil height. Measurements of nonlinear harmonic distortion, and the
compensation of this distortion by the nonlinear compensation algorithm developed in §3.3.7 are
presented.
Chapter 5
146
5.1. Linear equalisation
Perhaps the simplest linear equaliser for a loudspeaker is that for controlling the Q-value and cut-off
frequency described by Leach (1990). Leach describes an arrangement wherein the input to the power
amplifier is processed by a second-order active analogue filter. The filter provides a band-stop
characteristic described by the following s-domain transfer function:
H c ( s) =
s 2 ω 02 + s ω 0 Qtc + 1
s 2 ω d2 + s ω d Qd + 1
(5.1)
where Qd and ωd are the desired Q-value and cut-off frequency, respectively.
A discrete-time version of (5.1) may be derived by mapping the poles and zeros from the s-plane to
the z-plane according to the exponential mapping used in §2.3.2, producing
H c ( z) = σ c
1 + b1·a z −1 + b2·a z −2
1 + a1·d z −1 + a 2·d z − 2
(5.2)
The feedforward coefficients are given by
b1·a = a1
b 2· a = a 2
(5.3)
where a1 and a2 are the feedback coefficients of a discrete-time model of the loudspeaker, as described
in §2.3.4.
The feedback coefficients are determined by the desired cut-off frequency and damping factor
according to
a1·d = −2e −ωd ζd cos  ω d 1 − ζ 2d 


a 2· d = e
(5.4)
− 2 ωd ζ d
The overall system gain σc is chosen to provide unity above the cut-off frequency, and is thus given
by:
σc =
1 − a1·d + a 2·d
1 − b1·a + b2·a
(5.5)
5.1.1. Acoustic response with equalisation
An example of how this filter affects the loudspeaker’s frequency response, for three different filter
cut-off frequencies, is shown in Figure 5.1. In this figure, the frequency response of the loudspeaker is
synthesised from measured small-signal parameters. The overall Q-value of this loudspeaker (Qtc) is
about 5.0, typical for a ‘microspeaker’ placed in a small cavity. Parameters of the equaliser are chosen
to produce an equalised Q value of 1.0. The three cut-off frequencies chosen in the equaliser are one
half (½ ×) the same (1 ×) and twice (2 ×) the original resonance frequency of the loudspeaker.
0.8
0.6
40
0.4
20
100
1000
Fs/2
20
10
dB
147
1
60
Imaginary Axis
dB re 20µPa @ 1m / 1.0 V
Applications of active control of a loudspeaker
0.2
0
−0.2
−0.4
0
−0.6
−10
−0.8
−20
−1
−30
100
1000
Frequency (Hz)
Fs/2
−1
−0.5
0
Real Axis
0.5
1
Figure 5.1: Properties of the linear equaliser, for three different cut-off frequencies; UPPER LEFT: equalised
loudspeaker response (solid), unequalised (broken); LOWER LEFT: response of equaliser; RIGHT: pole-zero plot in zplane of equaliser’s transfer function. The response is shown for settings of the equaliser where the cut-off
frequency is set to ωd = 0.5 ω0 (top curve), ωd = 1.0 ω0 (middle curve), and ωd = 2 ω0 (bottom curve).
This type of equaliser is particularly useful for the microspeaker (Figure 2.1). As can be seen in
Figure 5.1, the un-equalised response of the microspeaker has a very high Q-value. This is due to the
low magnetic strength of the microspeaker, relative to other, typical loudspeakers. This high Q-value
is typically reduced by introducing some type of acoustic resistance, either to rear of the microspeaker,
or by a highly damped leak in the rear cavity. As shown Figure 5.1, this high Q-value can be
‘damped’ by the equaliser of (5.2).
One reason the equaliser of (5.2) is not commonly used is that the resonance frequency, ω0 in (5.2), is
subject to manufacturing tolerance and drift, as discussed in §2.4. As shown in (5.3), the loudspeakerdependent coefficients of the equaliser in (5.2) can be determined from the discrete time model of the
loudspeaker presented in §2.3.4. The techniques described in chapter 3 can adaptively identify these
parameters. Thus, the loudspeaker system identification techniques presented in chapter 3 can tune the
equaliser to the loudspeaker quite in a straightforward manner.
Note that when the filter cut-off frequency is chosen to be below the loudspeaker’s cut-off frequency,
the filter gain is positive below the cut-off frequency, and vice versa when the filter’s cut-off
frequency is above that of the loudspeaker. This has very important consequences for the variation of
the displacement response with respect to filter cut-off frequency, as discussed below.
Chapter 5
148
5.1.2. Displacement response with equalisation
An important property of the equaliser defined by (5.2) is its effect on the displacement frequency
response function. This effect is plotted in Figure 5.2, for the same values of the equaliser as plotted
in Figure 5.1.
10
dB re 1.0mmm / V
0
−10
−20
−30
−40
−50 2
10
3
10
Frequency (Hz)
4
10
Figure 5.2: Effect on the displacement response on the linear equaliser of (5.2). Broken: response without
equaliser; Solid: response with equaliser with ωd = 0.5 ω0 (top curve), ωd = 1.0 ω0 (middle curve), and ωd = 2
ω0 (bottom curve).
For the case where the cut-off frequency of the equaliser is the same as the loudspeaker’s resonance,
the peak value of the displacement response is reduced by over 10dB. This has an important
consequence on the maximum allowable input voltage. As discussed in §2.2.1, part B, the
microspeaker under consideration has a hard displacement limit at 0.35 mm (peak). From a systemlevel standpoint, this represents a limit that cannot be exceeded. As can be seen in the middle curve of
Figure 5.2, using the equaliser of (5.2) will increase the headroom by over 10dB.
It may also be possible to adaptively set the cut-off frequency of the equaliser in (5.2). As can be seen
for the different cut-off frequencies plotted in Figure 5.2, the peak value of the displacement response
reduces by 12dB per doubling of the cut-off frequency. It may be possible to use this effect as a
‘dynamic displacement limiter.’ Some success has been achieved with similar systems using analogue
processing (Bjerre, 1993). A digital implementation using this approach is considered a subject for
further research.
Applications of active control of a loudspeaker
149
5.2. Compensation of nonlinear distortion
As explained in other parts of this thesis, compensation of nonlinear distortion in loudspeakers by
electronic means has been a subject of research for several years. To date, it has been reasoned by
much of the loudspeaker industry that the added cost and expense of electronic distortion
compensation would not be reasonable. It has been considered that the cost of such systems would be
much larger than the proper mechanical construction to keep distortion within acceptable limits.
As mentioned in the introduction, it was suggested by Klippel (2000) that a distortion compensation
system can offer a net benefit to the loudspeaker’s construction. Specifically, it was suggested by
Klippel that the distortion caused by non-uniformity in the transduction coefficient (B·l-factor) in an
‘equal-hung’ voice-coil can be sufficiently compensated over the range of excursions of three-times
the coil height, with moderate increased output requirement from the amplifier.
Here an ‘equal-hung,’ or ‘equal-height,’ voice-coil, as shown in Figure 5.3, is one which has the same
height as the magnet gap. This is in contrast to an over-hung voice-coil, where the voice-coil’s height
is greater than the magnet gap’s height, or an ‘under-hung’ voice-coil, where the opposite is the case.
Over-hung voice coil
Equal-hung voice coil
Under-hung voice coil
Figure 5.3: Coil shapes for over-hung, equal-hung and under-hung voice-coil, as would be implemented in a
microspeaker.
As discussed in §2.1.5, a loudspeaker’s acoustic output1 is directly determined by its volume
acceleration. For a given acoustic output, if the effective diameter of a loudspeaker is halved, the axial
diaphragm vibration displacement must be increased by a factor of four. As there is typically
commercial pressure from product marketing requirements to minimise the overall diameter of a
loudspeaker, an important aspect of loudspeaker design is its maximum vibration displacement.
1
Acoustic output here denotes the pressure-distance product produced by a loudspeaker in free-field (in the
absence of any acoustically reflecting surfaces.)
150
Chapter 5
For the microspeaker (Figure 2.1) the coil mass dominates the total moving mass.1 As explained in
§2.1.7, the characteristic sensitivity (or system gain) of a loudspeaker is inversely proportional to the
total moving mass. By shortening the height of the voice-coil, the total moving mass may be reduced,
thereby increasing the loudspeaker’s sensitivity.
However, shortening the coil height has a problem. It will increase nonuniformity in the transduction
coefficient, thereby increasing nonlinear distortion. For this reason, loudspeaker designers commonly
use over- or under-hung voice-coils in high-displacement loudspeakers to keep nonlinear distortion at
acceptably low levels
From the perspective of loudspeaker optimisation, an over-hung voice-coil is sub-optimal, because it
requires excess diaphragm mass, reducing its sensitivity. Conversely, an under-hung voice-coil is a
sub-optimal design because it requires excess magnet material, which increases the loudspeaker’s
overall cost and weight. It is, therefore, considered here that the nonlinear distortion created by
nonuniformity in the transduction coefficient be compensated electronically.
Electronic compensation of nonlinear distortion does, of course, have its own cost. The cost is
measured in this study by the additional amplifier output required for compensation of nonlinear
distortion, measured in peak amplitude volts. As mentioned in the introduction, many audio products
already employ considerable digital signal processing. It is assumed that the processing needed for
distortion compensation is simple relative to existing signal processing algorithms. The cost of the
hardware needed for performing the processing for distortion compensation is, therefore, not
considered here.
A series of simulations have been made to find the optimal trade-off between increase in sensitivity
due to coil height reduction, and increase in amplifier output required for nonlinear distortion
compensation. Details of the simulation method are presented in §5.2.1 below, and the results and
discussion are presented in §5.2.2.
To test the hypothesis that a net benefit is derived from shortening the coil height and compensating
the resulting nonlinear distortion electronically, a set of special microspeakers have been prepared
with shortened coil heights. The simple nonlinear distortion compensation algorithm presented in
§3.3.7 has been used to compensate for the nonlinear distortion created by the increased nonuniformity
in the transduction coefficient caused by shortening the coil height. A set of measurements of
harmonic distortion of a narrow-band signal to asses the performance of the distortion compensation
algorithm are presented in §5.2.3.
5.2.1. Simulations of effective sensitivity increase vs. coil height
As discussed above, the additional amplifier output required for compensation of nonlinear distortion
generated by nonuniformity in the transduction coefficient has been simulated for a range of coil
heights, as a function of displacement (excursion) level. The range of voice coil heights considered is
from 0.1mm to 2.1mm, as per Figure 5.4.
1
The total moving mass mt, defined in (2.34), is the sum of the mass of the diaphragm, voice-coil, and mass-like
acoustic loading. The diaphragm of a microspeaker, made of thin polycarbonate plastic, has a mass of about
1mg. Mass-like acoustic loading on the diaphragm, under atmospheric conditions, is about 1mg. Depending
on loudspeaker type, the mass of the coil will be 50 – 100 mg.
Applications of active control of a loudspeaker
151
Figure 5.4: Variation of coil height made in simulations.
Lumped parameter quantities vs. coil height
Changing the coil height directly affects several ‘lumped parameters’ of the loudspeaker:
• nonuniformity in the transduction coefficient
• moving mass, md
• blocked (DC) electrical resistance Reb
Nonuniformity in the transduction coefficient is determined by variation in the magnetic field along
the coil gap B(x), and by the effective coil height in the gap, according to two different formulae
presented below. Data on variation in the magnetic field along the coil gap B(x) was derived from a
FEM simulation, which is plotted in Figure 5.5.
0.8
0.7
Magnetic field (Tesla)
0.6
0.5
0.4
0.3
0.2
0.1
0
−1
−0.5
0
0.5
Distance along magnetic gap (mm)
1
Figure 5.5: Magnetic field strength vs. distance along coil gap, from FEM simulation.
Changes to the force factor profile φ(x) due to changes in the coil height can be interpreted in two
ways. We can consider a coil length that varies directly with the coil height, and thus the force factor
profile is determined from
hu
∫
φ( x d ) = l h B (ξ − x d )dξ .
hl
(5.6)
Chapter 5
152
This produces force-factor profiles, for different coil heights, as shown in the upper half of Figure 5.6.
Alternatively, we can consider a fixed coil length, l0, which is independent of the coil height, and thus
determined from
φ( x d ) =
l0
hu − hl
hu
∫ B (ξ − x
d
)dξ .
(5.7)
hl
For different coil heights, this produces force-factor profiles as shown in the lower half of Figure 5.6.
Net force factor (N/A)
Length proportional to height
0.6
0.5
0.4
0.3
0.2
0.1
Net force factor (N/A)
0
−3
−2
−1
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
0.6
0.5
0.4
0.3
0.2
0
1
2
3
0
1
Distance [diaph. excur.] (mm)
2
3
Invariant Length
0.1
0
−3
−2
−1
Figure 5.6: Determination of force factor profile, φ(x). UPPER: Coil wire length proportional to coil height, as
per (5.6). LOWER: Coil wire length fixed, as per (5.7).
In order to have a fixed coil length for shorter coil heights, the coil would have to be thicker. A
thicker coil would require a wider magnet gap, which would reduce the magnetic field strength, and
consequently the sensitivity. Magnet gap widths and coil thicknesses are typically finely optimised.
Therefore, it has been chosen to study only the case of (5.6), where the coil length varies linearly with
the coil height.
The blocked electrical (DC) resistance is, for the simulation, determined according to the coil height by
Reb = R0 + Reff
=1+ 8
h
h0
h
(5.8)
1.2 × 10 −3
In (5.8), R0 describes a ‘residual’ resistance, in the output impedance of the amplifier, and the
electrical connections between the amplifier and the loudspeaker.
Applications of active control of a loudspeaker
153
The mass md has been determined according to the coil height according to.
m d = m a + meff
= 1.78 × 10
h
h0
−6
+ 18.5 × 10
−6
(5.9)
h
1.2 × 10 −3
In (5.9), ma is the effective mass presented by the acoustic load, which is obviously invariant to the
coil height. In the loudspeakers under study, it has been found that the mass of the diaphragm is
significantly less than 1mg. Consequently, the total effective moving mass md is simply the sum of the
coil mass and ‘acoustic’ mass.
For these simulations, it was decided to keep the resonance frequency fixed. The stiffness has
therefore been varied along with the mass, so as to have a constant resonance frequency, as so:
kd , 0 = md (2π f 0 )
2
(5.10)
The variation in various lumped parameter elements with the coil height used for simulations is shown
in Figure 5.7. Attention is drawn to the lower right chart of Figure 5.7, which shows the equivalent
rear-cabinet volume, VAS. For the shortest coil heights, and thus the lowest moving mass values, a
rather low stiffness (high compliance) is necessary to achieve the same f0. This produces significantly
higher VAS than would be typical for the loudspeaker under study (4cm3 is typical.)
Attention is also drawn to the lower left chart of Figure 5.7. The shortest coil heights produce a very
low DC resistance, and thus low Qtc. The shortest of coil heights produce a Qtc of around 0.3, which is
perhaps lower than could be thought ideal. This is shown in the small-signal far-field acousticpressure sensitivities, calculated from the lumped parameter quantities for the different coil heights,
shown in Figure 5.8.
md (MMD)
Re
15
Mass (mg)
Ohms
30
10
5
0
0
0.5
1
1.5
20
10
0
2
0
0.5
1
1.5
2
1.5
2
1
1.5
Coil Height
2
kd (1/C
phi (B•l)
)
MD
0.8
800
600
N/m
N/A
0.6
0.4
0.2
0
400
200
0
0.5
1
1.5
0
2
0
0.5
Qtc
1
VAS
30
3
Volume (cm )
2
Q
1.5
1
0.5
0
0
0.5
1
1.5
Coil Height
2
20
10
0
0
0.5
Figure 5.7: Variation in ‘lumped parameter’ quantities with coil height, for purposes of simulation.
Chapter 5
154
The linear pressure/voltage frequency response for the different coil heights was calculated according
(2.35) using the parameter value for different coil heights. The frequency response all of these coil
heights are plotted in Figure 5.8.
90
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
80
70
60
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
50
40
30
2
10
3
10
Frequency (Hz)
4
10
Figure 5.8: Small-signal far-field acoustic pressure sensitivity (referenced to 1m), for LPM quantities
calculated for the different coil heights.
Effective sensitivity vs. coil height and excursion
The effective sensitivity, for the combined controller-amplifier-loudspeaker system, has been
calculated to evaluate the trade-off between the increased sensitivity provided by a shorter height
voice-coil, and the decrease in sensitivity created by the additional amplifier output required for
compensation of the distortion generated by shortening the voice coil height. For the simulation
herein, the effective sensitivity Seff has been calculated as a function of excursion, based on the small
signal sensitivity S0, and the ‘correction gain’ at the specified displacement, Cdisp·n, according to:
S eff =
1 p1m ( s )
C c v c ( s)
(5.11)
x~0
The correction Cc gain represents the ratio of the peak value of the controller output u(t) to the peak
value of the controller input w(t), as per Figure 3.5.
Cc =
Pk { u (t )}
Pk { w(t )}
(5.12)
This effective sensitivity has been plotted as a function of peak excursion, for the above mentioned
range of coil heights, for several different frequencies. The results of the simulations are plotted in
Figure 5.9 - Figure 5.25. The results for each frequency are plotted in a separate figure.
Nonuniformity in the suspension stiffness k(xd) has not been considered in these simulations, as
measurements of its nonuniformity have shown it to be reasonably uniform. Simulations are made for
Applications of active control of a loudspeaker
155
excursions up to 2.0mm peak, which would cause the diaphragm-coil assembly to contact the magnet
or frame, occurring at about 0.35mm, as discussed in §2.2.1, part B. This effect is not considered in
the simulations results. As this results in an effective stiffness k(xd) that is infinite, an infinite
correction gain would be required to overcome this problem.
5.2.2. Simulation results
It can be seen in each of Figure 5.9 - Figure 5.25 that, at small displacements (less then 0.1mm Pk),
the shortest coil height has the highest effective sensitivity.
There seems to be some ‘critical displacement,’ below which the shorter voice coil heights have a
higher effective sensitivity, and above which the larger coil heights have a higher effective sensitivity.
The lower effective sensitivity for the shorter coil heights at higher displacement is due to the higher
correction gain, defined above (i.e. the additional amplifier output needed to compensate for the
resulting nonlinear distortion).
At lower frequencies this ‘critical displacement’ occurs at a single, well defined displacement level of
about 0.8mm Pk, for all of the coil heights. This is seen for all frequencies up to 633 Hz (plotted in
Figure 5.9 through Figure 5.14). This can be understood as the phenomenon occurring below the main
resonance frequency f0.
At frequencies around the main resonance frequency, this ‘critical displacement’ is not so well
defined. From about 825 Hz (plotted in Figure 5.15) through 1816 Hz (plotted in Figure 5.18), the
peak displacement level above which the larger voice-coil heights have a higher effective sensitivity is
different for different coil heights.
At higher frequencies, 2000Hz and above (plotted in Figure 5.19 through Figure 5.25), this critical
displacement is again well defined, occurring at a higher level of about 1.1mm.
From these simulations, it is concluded that a significant increase in sensitivity can be obtained by
shortening the voice-coil height. For the motor structure under consideration (i.e. the magnetic field
variation shown in Figure 5.5), it is concluded that shortening the coil height from 1.2mm to 0.3mm
should provide a sensitivity increase of about 8dB, when used over the specified displacement range of
0.35mm Pk. Furthermore, the additional amplifier output required for compensating the distortion
resulting from a reduction of the coil height from 1.2mm to 0.3mm should be marginal, i.e. less than
1dB.
Chapter 5
156
169Hz (0.2×f )
221Hz (0.3×f )
0
0
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
90
50
40
30
0
0.5
1
Pk Excursion (mm)
1.5
80
70
60
50
40
30
0
2
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
Figure 5.9: 168 Hz
0.5
0
80
70
60
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
50
40
1
Pk Excursion (mm)
1.5
70
60
50
40
30
0
2
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
Figure 5.11: 287 Hz
0.5
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
2
0
90
50
40
1
Pk Excursion (mm)
1.5
633Hz (0.8×f )
0
0.5
1
Pk Excursion (mm)
Figure 5.12: 374 Hz
487Hz (0.6×f )
30
0
2
374Hz (0.5×f )
0
0.5
1.5
Figure 5.10: 221 Hz
287Hz (0.3×f )
90
30
0
1
Pk Excursion (mm)
1.5
Figure 5.13: 487 Hz
2
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
50
40
30
0
0.5
1
Pk Excursion (mm)
1.5
Figure 5.14: 633 Hz
2
Applications of active control of a loudspeaker
825Hz (1.0×f )
1073Hz (1.3×f )
0
0
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
90
50
40
30
0
0.5
1
Pk Excursion (mm)
1.5
70
60
50
40
30
0
2
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
Figure 5.15: 825 Hz
0.5
70
60
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
80
50
40
1.5
2
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
50
40
30
0
Figure 5.17: 1397 Hz
0.5
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
2
0
90
50
40
1
Pk Excursion (mm)
1.5
3083Hz (3.7×f )
0
0.5
1
Pk Excursion (mm)
Figure 5.18: 1819 Hz
2368Hz (2.9×f )
30
0
2
0
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
1
Pk Excursion (mm)
1.5
1819Hz (2.2×f )
0
0.5
1
Pk Excursion (mm)
Figure 5.16: 1073 Hz
1397Hz (1.7×f )
90
30
0
157
1.5
Figure 5.19: 2368 Hz
2
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
50
40
30
0
0.5
1
Pk Excursion (mm)
1.5
Figure 5.20: 3083 Hz
2
Chapter 5
158
4013Hz (4.9×f )
5224Hz (6.3×f )
0
0
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
90
50
40
30
0
0.5
1
Pk Excursion (mm)
1.5
80
70
60
50
40
30
0
2
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
Figure 5.21: 4013 Hz
0.5
0
80
70
60
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
50
40
1
Pk Excursion (mm)
1.5
2
Figure 5.23: 6800 Hz
Sensitivity (dB re 20µPa @ 1m / 2.83 V)
90
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
50
40
0.5
1
Pk Excursion (mm)
1.5
Figure 5.25: 11523 Hz
0.1mm
0.3mm
0.5mm
0.7mm
0.9mm
1.1mm
1.3mm
1.5mm
1.7mm
1.9mm
2.1mm
80
70
60
50
40
30
0
0.5
1
Pk Excursion (mm)
1.5
Figure 5.24: 8852 Hz
11523Hz (14.0×f0)
30
0
2
8852Hz (10.8×f )
0
0.5
1.5
Figure 5.22: 5224 Hz
6800Hz (8.3×f )
90
30
0
1
Pk Excursion (mm)
2
2
Applications of active control of a loudspeaker
159
5.2.3. Distortion compensation on shortened- height voice-coil loudspeakers
In order to verify the simulation results presented in §5.2.2 above, a set of modified coil-height
loudspeakers was prepared on which measurements were made. The modified coil-height speakers
were made according to the specification shown in Figure 5.26. These modified coil-height
loudspeakers were prepared by Philips Speaker Solutions for the purpose of this research.
Voice Coil Height
Original Coil Height
1.2 mm
Plastic insert
Coil Height = 2 × Magnet Gap
Plastic insert
Coil Height = Magnet Gap
Plastic insert
Coil Height = ½ × Magnet Gap
0.8 mm
0.4 mm
0.2 mm
Figure 5.26: Specially prepared microspeakers with shortened coil heights.
Measurements of the linear response
Measurements of the linear frequency response of these modified coil height loudspeakers are shown
in Figure 5.27. Measurements were made according to the set-up described in §A.4 in Appendix A
(see p. 177.) The loudspeakers were placed in an acoustically free field. The loudspeakers were
mounted in a closed-box, with a rear cavity volume of approximately 2.0cc. In this way, the acoustic
environment corresponded as closely as possible to that assumed by (2.36).
Chapter 5
160
Acoustic pressure/voltage sensitivity
100
dB re 20µPa @ 0.28m / 1.0V in
90
80
70
60
50
1.2mm
0.8mm
0.4mm
0.2mm
40
300
400
500
600
800
1000
Frequency (Hz)
2000
3000
4000
6000
Figure 5.27: Linear pressure/voltage frequency response of specially prepared shortened-voice coil height
loudspeakers. The original specification voice coil height, 1.2mm, is shown in blue for reference.
The acoustic frequency response measurements in Figure 5.27 show that the shorter coil height
loudspeakers have a higher characteristic sensitivity. Recall that the characteristic sensitivity, S0
defines the sensitivity above the resonance frequency, as per (2.36). As can be seen in Figure 5.27, the
characteristic sensitivity of the shortest-height voice coil (0.2mm) is approximately 10dB above the
original-specification voice coil height (1.2mm).
The higher resonance frequency of the shorter-height voice-coil speakers is indicative of the lighter
mass produced by shortening the voice coil. As can be seen below the resonance frequency, all
speakers have roughly the same sensitivity. As, below resonance, the response is dominated by the
suspension stiffness, it can be said that the stiffness of all of the samples is roughly the same. The one
exception is the 0.8mm voice-coil height sample. By comparison to the response of the original
specification 1.2mm height voice coil, the 0.8mm high voice-coil sample has the same resonance
frequency, but has approximately 3dB higher voltage sensitivity throughout the measured frequency
range. This can be attributed to a slightly lower suspension stiffness in the 0.8mm high voice-coil
sample. Small deviations in the suspension stiffness from one sample to another are known to occur
due to manufacturing tolerances, as discussed in §2.4.
The nominal power sensitivity, calculated from the voltage sensitivity, is plotted in Figure 5.28. This
nominal power sensitivity has been calculated from the voltage sensitivity according to:
p1W ( s ) = p1V ( s ) Reb·n
(5.13)
where p1W(s) is the nominal power sensitivity, p1V(s) is the voltage sensitivity, and Reb·n is the DCresistance of the sample concerned.
Applications of active control of a loudspeaker
161
Acoustic pressure, nominal power sensitivity
100
dB re 20µPa @ 0.28m re sqrt(Reb) V in
90
80
70
60
50
1.2mm
0.8mm
0.4mm
0.2mm
40
300
400
500
600
800
1000
Frequency (Hz)
2000
3000
4000
6000
Figure 5.28: Nominal pressure / power sensitivity for the same loudspeakers for which measurements are
plotted in Figure 5.27. This represents the theoretical 1W sensitivity.
The increase in nominal power sensitivity for the 0.2mm high coil is only about 4dB over the original
1.2mm coil height. This is due to the lower DC resistance of the 0.2mm high coil (approx. 1.5Ω)
compared to the 1.2mm high coil (7.2Ω). The nominal power sensitivity below the resonance
frequency is somewhat lower for the shorter-height voice coils. It is expected that this is due to a
higher effective stiffness in the shorter-height voice coil speakers - an accidental result not caused by
shortening the voice coil height, but due to manufacturing tolerances. This can be seen in the lower
voltage sensitivity that the shorter voice-coil height speakers have below the resonance frequency
shown in Figure 5.27. With the below-resonance power sensitivity given by
p1W ( s )
f << f 0
= s2
S d φ0
Reb
(5.14)
It is expected that if the effective stiffness were to have been the same for all for loudspeakers, the
below-resonance power sensitivity would be more similar between them.
In the simulations presented in §5.2.2 above, the suspension stiffness was changed for different coil
heights, so as to keep the resonance frequency constant. This was not done in the physically prepared
samples. Although it is generally possible to freely adjust the suspension stiffness, by adjusting the
thickness of the plastic making up the suspension material, this was not done, in order to simplify the
sample preparation process.
Set-up and parameter tuning
The distortion compensation algorithm developed in §3.3.7 is used to compensate for nonlinear
distortion created in the modified coil-height loudspeakers. Harmonic distortion is assessed with a
162
Chapter 5
specially synthesised narrow-band signal. This signal is synthesised by band-pass filtering white noise
with 10th-order Chebyshev filters, providing a 30Hz pass-band with high out-of-band cut-off slope.
This signal was chosen to assess the harmonic distortion created by the loudspeaker with a simple
measurement of the autopower spectrum of the acoustic pressure field. This narrow-band noise signal
was chosen over sinusoidal signals to demonstrate the proper operation of the distortion compensation
algorithm. It was considered that this narrow-band noise signal would be a more effective assessment
of the operation of the distortion compensation algorithm than more commonly used sinusoidal
signals, as the harmonic distortion created by sinusoidal signals could be compensated for by simple
harmonic superposition.
For all measurements of the compensation of nonlinear distortion, the input level to the loudspeaker
was set so that the maximum allowable diaphragm-coil displacement was achieved. For the
loudspeakers under investigation here, this was 0.35mm Pk. This limit is set by the mechanical
construction of this particular loudspeaker type.1
Parameters of the loudspeaker model were obtained in the following manner. The effective moving
mass of the loudspeaker was determined using the method described in Appendix B. The remaining
parameters describing the linear characteristics were determined using the system identification
algorithms presented in chapter 4, above. Specifically, the electrical current output error form
described in §4.2 was used. Parameters were allowed to converge to their final values using a white
noise signal.
Parameters describing nonlinear characteristics were determined by trial-and-error, tuning φk (the
coefficients of a polynomial representation of φ(xd) ) by hand. Values of φk were chosen which
minimised the harmonic distortion over a range of frequencies of the narrow-band signal.
The commercial instrument discussed in §2.2.1, part A , which can determine φk, though available at
the time measurements were made, could not be used. This is due to the fact that at that time the
commercial system could not analyse loudspeakers with a resonance frequency higher than 800Hz.
As can be seen from Figure 5.27, the resonance frequency of these loudspeakers is above this limit.
The values of φk determined by hand compared favourably with those determined by this commercial
instrument on similar loudspeakers with a resonance frequency lower than 800Hz.
The set-up of the loudspeakers under test was otherwise the same as that for which the linear response
of the shortened-voice-coil-height loudspeakers was measured, as described above.
Measurement results
Harmonic distortion of the narrow-band signal can be seen in measurements of the autopower
spectrum of the acoustic pressure, shown on the left side of Figures 5.29 through 5.32. The reduction
in the harmonic distortion provided by the nonlinear control algorithm in (3.28) is plotted in the same
figures. In each of these figures, the solid line shows the spectra with the compensation on, and the
broken line shows the spectra with the compensation off.
The autopower spectrum of the voltage from the controller to the speaker is shown on the right side of
of Figures 5.29 through 5.32. Again, in these figures, the solid line shows the spectra with the
compensation on, and the broken line shows the spectra with the compensation off. As can be seen in
these figures, there is an increase in the ‘harmonics’ of the narrowband signal when the compensation
1
As discussed in §2.2.1, part B, higher displacements than 0.35mm peak cause contact between its diaphragmcoil assembly and magnet-frame. Therefore this hard limit could not be exceeded in these measurements.
Applications of active control of a loudspeaker
163
is on. This represents the ‘anti-distortion’ generated by the nonlinear control algorithm needed to
compensate for the distortion generated in the loudspeaker.
The harmonic distortion of the narrow band signal generated by each of the different coil-height
loudspeakers is summarised in Table 5.1. The percentage-values shown in this table are sum of the
level of 2nd-4th harmonics relative to the fundamental. The percentage harmonic distortion, calculated
in this way, is shown in Table 5.1 for both the compensation on and off.
Coil height
0.8mm
0.4mm
0.2mm
0.2mm
Fig. No.
Figure 5.29
Figure 5.30
Figure 5.31
Figure 5.32
BP sig.
Cen. Freq.
800 Hz
800 Hz
800 Hz
1200 Hz
Max % Harm. Dist.
Comp. Off Comp. On
5.4 %
12.9 %
17.8 %
3.1%
1.2 %
3.2 %
3.2%
2.3%
Table 5.1: Reduction of harmonic distortion for each coil-height.
As can be seen in Table 5.1, the shorter coil-height loudspeakers generate more harmonic distortion;
the 0.8mm height coil (the highest) shows the least distortion, and the 0.2mm height coil (the shortest)
shows the most increase in harmonic distortion. This is as expected, due to the increase in
nonuniformity of the transduction coefficient caused by decreasing the coil height.
The nonlinear control algorithm was found to reduce the maximum harmonic distortion from at most
17.8% (in the 0.2mm height coil) to about 3%. The highest amount of harmonic distortion measured
with the compensation on was 3.2% (for the same 0.2mm height coil).
The harmonic distortion present in the original-specification height coil (1.2mm) was about 5% for the
same signal, at the same displacement. It is concluded, therefore, that the nonlinear control algorithm
of (3.28) is capable of compensating the distortion caused by the reduction in coil height, at least for
this narrow-band signal used for these measurements.
The additional voltage needed for compensation of the resulting distortion was at most 10% above the
voltage level with the compensation off. This translates to an increase of about 0.85dB. These
measurements thus seem to confirm the conclusions drawn from the simulations presented in §5.2.2.
Specifically, these measurement results seem to confirm the conclusion that reducing the coil height
from 1.2mm to 0.3mm would result in an 8dB voltage sensitivity increase, and require at most 1dB of
additional amplifier output for compensation of the resulting distortion. In measurements, the coil
height was reduced to 0.2mm, the voltage sensitivity increase was found to be 10dB, and the
additional amplifier output required for compensation was at most 0.85dB.
Chapter 5
164
Acoustic Pressure at 28cm − Solid: Active; Broken: Inactive
Input Voltage − Solid: Active; Broken: Inactive
70
−10
60
dB re 1.0Volt, per 2.5Hz bin
dB re 20µPa, per 2.5Hz bin
−20
50
40
30
20
10
−30
−40
−50
−60
−70
0
−80
−10
−90
−20
400
1000
2000
Frequency (Hz)
4000
400
1000
2000
Frequency (Hz)
4000
Figure 5.29: 0.8mm coil height, distortion analysis for NB signal centred on 800Hz. LEFT: Acoustic pressure;
RIGHT: Output voltage from amplifier.
Acoustic Pressure at 28cm − Solid: Active; Broken: Inactive
Input Voltage − Solid: Active; Broken: Inactive
70
−10
60
dB re 1.0Volt, per 2.5Hz bin
dB re 20µPa, per 2.5Hz bin
−20
50
40
30
20
10
−30
−40
−50
−60
−70
0
−80
−10
−90
−20
400
1000
2000
Frequency (Hz)
4000
400
1000
2000
Frequency (Hz)
4000
Figure 5.30: 0.4mm coil height, distortion analysis for NB signal centred on 800Hz. LEFT: Acoustic pressure;
RIGHT: Output voltage from amplifier.
Acoustic Pressure at 28cm − Solid: Active; Broken: Inactive
Input Voltage − Solid: Active; Broken: Inactive
70
−10
60
dB re 1.0Volt, per 2.5Hz bin
dB re 20µPa, per 2.5Hz bin
−20
50
40
30
20
10
−30
−40
−50
−60
−70
0
−80
−10
−90
−20
400
1000
2000
Frequency (Hz)
4000
400
1000
2000
Frequency (Hz)
4000
Figure 5.31: 0.2mm coil height, distortion analysis for NB signal centred on 800Hz. LEFT: Acoustic pressure;
RIGHT: Output voltage from amplifier.
Applications of active control of a loudspeaker
Acoustic Pressure at 28cm − Solid: Active; Broken: Inactive
165
Input Voltage − Solid: Active; Broken: Inactive
70
−10
60
dB re 1.0Volt, per 2.5Hz bin
dB re 20µPa, per 2.5Hz bin
−20
50
40
30
20
10
−30
−40
−50
−60
−70
0
−80
−10
−90
−20
400
1000
2000
Frequency (Hz)
4000
400
1000
2000
Frequency (Hz)
4000
Figure 5.32: 0.2mm coil height, distortion analysis for NB signal centred on 1200Hz. LEFT: Acoustic pressure;
RIGHT: Output voltage from amplifier.
166
Chapter 5
5.3. References
Bjerre, Egon, “Loudspeaker Arrangement with Frequency Dependent Amplitude Regulation,” U. S.
Patent 5,481,617. Filed: Mar. 1, 1993. Granted: Jan. 2, 1996.Assignee: Bang & Olufsen A/S,
Struer, Denmark.
Klippel, Wolfgang, “Direct Feedback Linearisation of Nonlinear Loudspeaker Systems,” J. Audio
Eng. Soc., 46, pp. 499-507. (Jun. 1998)
Klippel, Wolfgang J., “Adaptive Nonlinear Control of Loudspeaker Systems,” J. Audio Eng. Soc. 26,
pp. 939-954. (Nov. 1998)
Klippel, Wolfgang J., Personal, unwritten correspondence, Paris, France. (February 22, 2000)
Leach, W. Marshall , Jr., “A Generalised Active Equaliser for Closed-Box Loudspeaker,” J. Audio
Eng. Soc., 38, pp. 142-146. (Mar. 1990)
Schurer, Hans, Linearisation of Electroacoustic Transducers, Ph.D. Thesis, University of Twente
Enschede, ISBN 90-365-1032-5. (1997)
Schurer, Hans, Cornelis H. Slump, and Otto E. Herrmann, “Theoretical and Experimental Comparison
of Three Methods for Compensation of Electrodynamic Transducer Nonlinearity,” J. Audio Eng.
Soc., 46, pp. 723 – 740. (Sept. 1998)
167
6.
Conclusions
This thesis studies practical implications of applying active control to a loudspeaker. By developing a
discrete-time model of the continuous-time loudspeaker dynamics, digital processing for active control
is made sufficiently simple so that it may be cost-effective to introduce active control to existing
products. The adaptive feedforward architecture is found to be the most practical implementation of
active control. It is shown that active control can provide a net benefit to the loudspeaker, using both
linear and nonlinear processing.
A discrete-time model of a loudspeaker is developed in this thesis. This discrete-time model
simplifies the digital signal processing algorithms for active control. The key simplification is derived
from digital filters designed to have the same response characteristics as dynamics of the loudspeaker.
Simplicity is enabled by keeping the order of the digital filters the same as the order of dynamics of
the loudspeaker. This achieves simplification over previously published algorithms for active control,
which simulated continuous-time dynamics by numerical integration, or by high-order non-recursive
filters.
The adaptive feedforward architecture, using digital signal processing, is found to be the most
practical method for implementing active control of loudspeakers. Two alternative architectures,
feedback processing and non-adaptive feedforward processing, are considered impractical. The
feedback architecture is discarded due to its need for a direct feedback signal from the loudspeaker,
which is impractical or expensive to obtain. The pure feedforward (non-adaptive) architecture is
discarded due to its sensitivity to misalignment to the loudspeaker. The adaptive feedforward
architecture is chosen due to its ability to tune itself to changes in the loudspeaker caused by
temperature fluctuations and ageing. Furthermore, it is shown that the adaptive process, performed by
system identification, can operate effectively using a voice-coil current signal, which is simple and
inexpensive to obtain from the loudspeaker.
Active control is shown to provide a net benefit to a loudspeaker system. Standard linear active
control, or equalisation, enables certain parts of a loudspeaker’s response to be controlled
electronically. It is specifically shown that the highly undamped nature of small loudspeakers can be
controlled by simple equalisation. The adaptive feedforward architecture for active control, described
above, ensures the equaliser is properly tuned to the loudspeaker.
Active control is also shown to provide a net benefit through nonlinear processing. Nonlinear active
control can correct small nonlinearities introduced by relaxing certain standard loudspeaker design
requirements. Specifically, shortening the height of a loudspeaker’s voice-coil increases its sensitivity,
though introduces a certain amount of weak nonlinearity. Simulations and measurement show that
this weak nonlinearity can be compensated by active control. Furthermore, it is shown that the
additional amplifier voltage output required for compensation of this nonlinearity is less than the
increase in sensitivity provided by shortening the voice-coil height. Simulations and measurements
show this method can provide a net increase in voltage sensitivity up to 9dB, and a power sensitivity
increase up to 4dB, over existing loudspeaker designs.
Nonlinear active control is assessed in this thesis by its ability to control harmonic distortion.
Intermodulation distortion – generally considered more important in regard to subjective loudspeaker
quality – has not been studied. Subjective tests assessing the performance of the nonlinear distortion
compensation algorithm have not been carried out either. It is expected that by reducing harmonic
distortion at the frequencies concerned, the algorithm would reduce intermodulation distortion by
consequence. This has not, however, been verified, and would be a suitable subject of further
investigation. Furthermore, it is recommended that some degree of formal subjective evaluation of the
168
Conclusions
nonlinear distortion compensation algorithm be performed before this algorithm is used in a
commercial product.
Results from all algorithms are from implementations using double-precision floating point
processing. Implementation of any of these algorithms in actual commercial products would be done
using fixed-point arithmetic. No investigations have been made on the impact of using fixed-point
arithmetic on these algorithms. Experience with other algorithms suggests that the most likely
problem in fixed-point implementation is the proximity of poles in various systems’ transfer functions
to the unit circle. This most likely will require 16 bit data input-output streams to be computed with
32 bit or 40 bit arithmetic. This kind of architecture has been successfully used in many audio DSP
applications. Thus fixed-point considerations for these algorithms are currently thought to be more a
matter of engineering implementation than further academic research.
169
Appendix A. Experimental set-up and tuning
The various experimental set-ups and hardware used through this thesis are presented in this appendix.
This includes set-ups for primary experiments presented in the thesis, as well as those used in
‘background research.’
A.1. Hardware implementation of algorithms
The algorithms for system identification of the loudspeaker presented in Chapter 4 and the distortion
compensation algorithm presented in §3.3.7 were written and implemented on a standard desktop PC.
The algorithms were written in ANSI C, using double-precision floating point arithmetic. Analogue
input-output was handled by a standard sound card.
The system identification algorithm developed in §4.2, i.e. the electrical current output error plant
model form, was used to identify those parameters of the loudspeaker which were known to change.
These identified parameters were then used in the linear dynamics and inverse dynamics blocks of the
nonlinear feedforward processor. A block diagram of the complete system architecture is shown in As
mentioned in §4.1.2.
A.2. Electrical impedance measurement
A.2.1. Differential vs. single-ended considerations
The simplest conceptualisation of the arrangement for using a shunt resistor to measure the electrical
current is shown in Figure A.1.
i(s)
vin(s)
Zunknown
Rshunt
vR·shunt(s)
vZ·unknown(s)
Figure A.1: Circuit arrangement for measurement of electrical current and voltage. Note this arrangement is
suitable only for differential inputs and outputs.
With this arrangement, the electrical current can be determined from the measured voltage vR·shunt(s)
according to
i( s) =
v R·shunt ( s )
R shunt
(A.1)
Experimental set-up and tuning
170
The electrical impedance may, therefore, be determined by
Z e·meas ( s ) = R shunt
v Z ·unknown ( s )
v R ·shunt ( s )
(A.2)
Note that in this arrangement, the ground of the two measured signals, vR·shunt(s) and vZ·unknown(s) are
different. Simultaneous measurement of current and voltage in this way will, therefore, require
differential analogue inputs on the analyser.
If only single-ended inputs are available, it will be necessary to reverse the polarity of the
measurement of the shunt resistance voltage. The negative value of Rshunt should consequently be used
as a calibration value to vR·shunt(s).
In some applications, all output as well as input will be single ended. This is generally the case if both
analogue input and output are handled by a standard PC computer sound card. In this case, the
arrangement shown in Figure A.2 must be used.
i(s)
vin(s)
Zunknown
Rshunt
vZ·1(s)
vR·shunt(s)
Figure A.2: Arrangement for measurement of electrical current and voltage with single-ended circuit inputs
and outputs.
The signals obtained from the arrangement in Figure A.2 must be interpreted somewhat differently
from those measured according to Figure A.1. The signal vR·shunt will still give a signal directly
proportional to the electrical current i(s). However, the signal vZ·1(s) will provide the voltage drop
across the unknown load and the shunt resistor. In this case, the electrical impedance of the unknown
load may be obtained from the measured signals as so:
Z e·meas ( s ) = R shunt
v Z ·1 ( s )
− R shunt
v R· shunt ( s )
(A.3)
A.2.2. Extraneous resistances
The arrangements for measuring electrical impedance shown in Figure A.1 and Figure A.2 assume
ideal, zero-resistance contacts and interconnect cables. In practice, contacts and interconnect cables
will have some finite resistance. It was found that, to ensure the accuracy of the displacement
predicted by the linear dynamics block in Figure 4.1, it was necessary to include the effect of the
output resistance of the power amplifier, and the resistance of the wires connecting the power
amplifier to the loudspeaker. This is due to the generally low impedance of electrodynamic
Appendix A
171
loudspeakers. A technique for determining these resistances and the values determined are described
in this section.
Extraneous resistance in the circuit may be considered in two parts:
• The output resistance from the power amplifier
• The contact and interconnect resistance between the voltage drop measurement point and that
point for which the impedance of the device-under-test is defined
These two resistances are represented as Rout and Rlead, respectively, in the circuit in Figure A.3.
Rout
i (s )
Rlead
vin(s)
Zunknown
Rshunt
vR·shunt(s)
vZ·2(s)
Figure A.3: Arrangement for measuring electrical impedance, with consideration of amplifier output resistance
and contact and interconnect resistance. This arrangement is only suitable for differential inputs.
The first of these, the output resistance from the power amplifier Rout, need not be considered in
ordinary measurement of the electrical impedance. It need be known only when is necessary to
predict the voltage drop across the device under test directly from the input to the power amplifier, as
is the case in the applications of feedforward control described in the main body of this thesis.
The second of these, the contact and interconnect resistance Rlead, does sometimes need to be
considered when measuring the electrical impedance. Typical values of Rlead are between 0.005 and
0.3 Ω, depending on the interconnect lead wire and the contact type. As the typical impedance of
loudspeakers is between 4 and 32Ω, this resistance needs to be accounted for if the electrical of the
loudspeaker needs to be measured with high absolute accuracy. If this is case, this resistance should be
determined separately, by e.g. shorting the terminals on the loudspeakers. With Rlead then known, the
measured impedance can be corrected according to
Z e·meas ( s ) = R shunt
v Z ·2 ( s )
− Rlead
v R· shunt ( s )
(A.4)
The correction according to (A.4) is generally applicable, as the capacitive and inductive components
of the interconnect cable and contacts will be multiple orders of magnitude below those of an
electrodynamic loudspeaker.
171
Experimental set-up and tuning
172
A.2.3. Determination of extraneous resistances with single-ended I/O
For single-ended input-output systems, it is necessary to determine both the lead and the amplifier
output resistance. A circuit showing the elements under consideration is shown in Figure A.4. Notice
that in this figure the unknown impedance has been replaced with a known resistance Rload.
i(s)
Rout
Rlead
vin(s)
vZ·1(s)
Rload [n]
Rshunt
vR·shunt(s)
Figure A.4: Arrangement for measurement of electrical current and voltage with single-ended circuit inputs
and outputs, with lead and amplifier output resistance considered.
A set of different load impedances, Rload[n] for n = 1 to N, are inserted into the circuit where the
unknown impedance would be located. For each Rload[n], the voltage across the shunt resistor vi·n and
the voltage across all components vZ·I are measured.
The relationship between these voltages and resistances is given by
R shunt v Z · I [n] = v R· shunt [n] (Rlead + Rload [n] + R shunt )
(A.5)
An error function may be determined from this
χ (Rlead , R shunt ) =
2
N
∑
n =1


 v R ·shunt [n](Rlead + Rload [n] + R shunt ) − v Z · I [ n]R shunt 


2
(A.6)
The partial derivatives of the error function with respect to Rlead and Rshunt lead to two equations. The
terms Rlead and Rshunt may be determined by setting these equations to zero, and solving the resulting
system. This may be expressed in the following matrix form:

Σ(v i [n] − v t [n])2 
 Rlead  Σn v i [n](v i [n] − v t [n])
n

R
=
2

Σ
Σ
−
(
)
v
[
n
]
v
[
n
]
v
[
n
]
v
[
n
]
 shunt 
i
i
i
t
n
n


−1
− Σ Rload [n]v i [n](v i [n] − v t [n])
 n

− Σ Rload [n]v i2 [n]


n


(A.7)
Now consider
vin = ic [ n](Rout + Rt [n])
(A.8)
where ic[n] is the current through the circuit i(s) for the nth value of Rload , and Rt[n] is the sum of Rlead,
Rshunt, and Rload[n]. Using the results above, the current for the nth value of Rload is given by
i m [ n] =
v R· shunt [n]
Rshunt
(A.9)
Appendix A
173
An error function is defined from (A.8) as so:
(
N
) ∑ (i [n](R
χ 2 Rout , v gen =
c
out
+ Rt [n]) − vin )
2
(A.10)
n =1
As above, the partial derivatives of this error function are taken with respect to Rout and vin. This
produces a set of two equations, which in matrix form are
2
 R out   Σ i cm [n] − Σ i cm [n]
n
n


v  =
i
[
n
]
N
−
Σ


gen
cm

  n

−1
2
− Σ i cm
[ n]Rt 
 n

 Σ i cm [n]Rt 
 n

(A.11)
Use of appropriate different values of Rload and the use of (A.11) with measured data permit
calculation of the output impedance of the amplifier (including cabling) as well as the ‘original’ output
voltage. The values shown in Table A.1 have been determined with this method for the experimental
equipment shown in Figure A.6 - Figure A.9.
Parameter
Rlead
Rshunt
Rout
vgen
Determined value
0.11981 Ω
1.0339 Ω
0.09493 Ω
0.951Vrms
Table A.1: Typical values of the interconnect cable and contact resistance (Rlead), the shunt resistance (Rshunt),
and the amplifier output resistance (Rout)
A.3. Experimental set-up for diaphragm for vibration measurement
The measurement set-up for measurement of parameters of a single-degree-of-freedom model of an
electrodynamic loudspeaker is shown in a block diagram in Figure A.5. Photographs of the test-bench
are shown in Figure A.6 and Figure A.7. These photographs are taken from the same equipment as
shown in Figure 4.1.
173
Experimental set-up and tuning
174
Speaker under test
Vacuum chamber
Transmitted & reflected
laser beam
Optical sensor head
Transparent window
Deflection mirrors
(scanner)
Vacuum
pump
+
-
Control cable
Angle
Control
Controller
Analogue velocity
signal
Voltage
Shunt
resistor
1
Current
Signal
Conditioning
& Digitisation
D/A
Power amplifier
Processing, Analysis,
Geometry data,
Signal generation,
and Storage
Figure A.5: Experimental set-up for measurement of parameters of a single-degree-of-freedom model for a
loudspeaker.
The laser vibrometer used was a commercially available device from Polytec, model OFV 303 sensor
head with companion OFV 3001 controller. Signal conditioning and digitisation of the analogue
signals was performed by an HP E1433A data acquisition card. Signals were analysed and stored
using the Cada-x Fourier Monitor software from LMS. A standard audio constant output-voltage
power amplifier was used to drive the loudspeaker. No other special equipment was used.
Appendix A
175
Figure A.6: Photograph of test-bench for the experimental set-up shown in Figure A.5, showing optical sensor
head, with scanning mirrors, for direct measurement of a loudspeaker. The measurement is being made inair, and thus the vacuum chamber is not present.
175
Experimental set-up and tuning
176
Input to data
acquisition
Signal processing & demodulation
electronics
Optical sensor head
Vacuum pump
Power amplifier
Coffee cup
Vacuum chamber
Shunt resistor for current measurement
Figure A.7: Photograph of the test bench for a variant of the set-up shown in Figure A.5, showing the vacuum
chamber & pump, used to remove the effects of acoustic loading from the loudspeaker. The vibration of the
loudspeaker can be measured in a vacuum by the laser vibrometer by virtue of a transparent window in the
vacuum chamber.
Appendix A
177
A.4. Experimental set-up for acoustic measurement
The experimental set-up for measurement of various aspects of the acoustic response of loudspeakers
described in different parts was as described in this section.
All acoustic measurements were performed in an anechoic chamber. The loudspeaker was mounted in
a variable back-volume test jig, and its acoustic response was measured with a standard laboratory
microphone. The distance from the loudspeaker to the microphone was 28cm, as in Figure A.8.
A photograph of equipment for signal generation, active control, and signal analysis in a control room
adjacent to the anechoic room is shown in Figure A.9
28cm
Measurement microphone
Loudspeaker, in jig with
variable rear-cavity volume
Anechoic room at Nokia Research Center,
Tampere, Finland
4.2m wedge tip - wedge tip
80Hz lower cut-ff
Figure A.8: Photograph of measurement set-up in anechoic chamber for acoustic measurements.
177
Experimental set-up and tuning
178
Shunt resistor for current measurement
Microphone signal conditioning
Power amplifier
Computer for real-time processing
Calibration resistor
Duct tape
Loudspeaker Voltage &
Current signal conditioning
Figure A.9: Photograph of experimental equipment used for acoustic measurement (located in room adjacent
to anechoic chamber.)
179
Appendix B. Experimental determination loudspeaker parameters
Classical theory for determination of parameters for loudspeakers is developed from the 1930’s, when
RMS voltmeters were the most sophisticated instrument available for analysing dynamic systems. This
technique, well-known to most electroacousticians, is described in Beranek’s Acoustics (1957) and the
well-known paper by Thiele (1961). The essence of the technique is to work out the loudspeaker’s
parameters from changes in its electrical impedance caused by known changes applied to its
mechanical load impedance. In Beranek’s case this is done by applying a known mass to the
diaphragm, and in Thiele’s case this is done by changing the volume of a sealed enclosure onto which
the loudspeaker is mounted.
In this dissertation, an unpublished method using an FFT analyser and laser Doppler vibrometer has
been used. Several methods using an FFT analyser have been published (Struck, 1987.) One method
using an FFT analyser and a laser vibration sensor has also been published (Monero, 1991). As the
technique used in this thesis is different from these previous techniques, and as it has not been
published elsewhere, it is presented in this appendix.
B.1. Parameters to be determined
The six basic parameters governing the loudspeaker’s electrical and mechanical behaviour can be
determined in two separate stages:
1. Determination of the electrical parameters: Reb, Leb, and φ0.
2. Determination of the mechanical parameters: md, cd, and kd.
The fitting of curves can be weighted according the standard deviation of the measured FRF functions.
The standard deviation can be estimated from the coherence.
B.2. Electrical parameter determination (Reb, Leb, and φ0)
An appropriate equation for determining the electrical parameters Reb, Leb, and φ0 comes from the coil
voltage equation. Dividing by the current ic(s) and subtracting the left-hand-side, we have
Reb + sLeb + φ0
ud ( s ) vc ( s )
−
= 0.
ic ( s ) ic ( s )
(B.1)
The parameters Reb, Leb, and φ0 can be determined by measuring the two frequency response functions
(FRF’s) u ( s) i( s) and v( s) i ( s) , then minimising the left hand side of (B.1) by a least-mean-square
method. To ease in notation, recalling that s = – iω, we define:
Yi ( f n ) =
ud ( −iωn )
ic (−iωn )
(B.2)
Ze ( fn ) =
vc (−iωn )
ic (−iωn )
(B.3)
With this notation, Yi (fn) is the complex-valued FRF of velocity / current at the frequency fn. Similarly,
Ze(fn) is the complex-valued electrical impedance at frequency fn. Substituting these expressions into
(B.1), one may define an error function to be minimised is as so:
χ 2 ( Reb , Leb , φ0 ) =
N stop
∑R
eb
n = N start
− iωn Leb + φ0Yi ( f n ) − Z e ( f n )
2
(B.4)
Appendix B
180
where Nstart and Nstop are the indices to fn giving the beginning and end of the frequency range over
which the parameters Reb, Leb and φ0 are to be fit. By taking the partial derivatives of χ2(Reb, Leb, φ0)
with respect to Reb, Leb, and φ0 and setting each to zero, a linear set of three equations is created as so:
N stop
∂χ 2
=
2 Reb + 2φ 0 Re{Yi ( f n )} − 2 Re{Z e ( f n )}
∂R n = N
∑
(B.5)
start
N
stop
∂χ 2
=
2ω2n Leb − 2ωnφ0 Im{Yi ( f n )} + 2ωn Im{Z e ( f n )}
∂Leb n = N start
∑
∂χ 2
=
∂φ0
N stop
∑ 2R
eb
(B.6)
Re{Yi ( f n )} − 2ωn Leb Im{Yi ( f n )} + 2φ0 Yi ( f n ) − 2[Re{Yi ( f n )}Re{Z e ( f n )} − 2 Im{Yi ( f n )}Im{Z e ( f n )}]
2
n = N start
(B.7)
Rewriting Eqs. (B.5)-(B.7) into matrix form provides:

0
Σ Re{Yi ( f n )}   Reb  
Σ Re{Z e ( f n )}
N


  

2
0
Σω n
− Σωn Im{Yi ( f n )}  Leb  = 
− Σωn Im{Z e ( f n )}


2
Σ Re{Yi ( f n )} − Σωn Im{Yi ( f n )}
  φ0  Σ[Re{Yi ( f n )}Re{Z e ( f n )} + Im{Yi ( f n )}Im{Z e ( f n )}]
Σ Yi ( f n )


(B.8)
where N is the total number of points over which the summation is made, given by Nstop – Nstart +1.
The parameters Reb, Leb, and φ0 are determined by multiplying the vector on the right-hand-side of Eq
(B.8) with the inverse of the matrix on the left-hand-side of (B.8).
The determined parameters are verified by plotting the terms in (B.1) against each other. The
determined electrical resistance Reb is verified by comparing
?
 v ( s)
u ( s) 
Reb = Re c
− φ0 d 
ic ( s ) 
 ic ( s )
(B.9)
Although the comparison requires a value for φ0, an incorrect value thereof would not mislead one
when making a comparison of plots of the right and left-hand side of (B.9). The determined value of
Leb is verified by comparing
?
 v ( s)
u ( s) 
− ωLeb = Im c
− φ0 d 
ic ( s ) 
 ic ( s )
(B.10)
Experimental determination loudspeaker parameters
181
Ohms (resistive)
8
6
4
2
0
400
600
800
1000
1200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
2000
1400
1600
1800
2000
Ohms (reactive)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure B.1: Verification of Reb (upper) and Leb (lower); in the upper frame, the grey1 trace shows the RHS of
(B.9), and the solid black trance shows the value Reb fit from this data; in the lower frame, the grey trace
shows the RHS of (B.10), and the solid black trance shows the value Leb fit from this data
The value of φ0 may be verified by comparing
φ0
u d ( s) ? v c ( s)
=
− Reb − sLeb .
ic ( s ) ic ( s )
(B.11)
An example of a comparison of the LHS and RHS of this equation is shown in Figure B.2. The real part
of this equation is shown in the upper frame of Figure B.2, and the imaginary part in the lower frame.
1
On colour print and electronic media, this will appear as a ‘cyan’ or ‘sky-blue’ trace in these figures.
Appendix B
182
15
10
5
0
400
600
800
1000
1200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
2000
1400
1600
1800
2000
5
0
−5
−10
Figure B.2: Verification of φ0; the upper frame shows the real part of (B.11), the LHS of which is plotted as a
solid black trace, and the RHS of which is plotted as a grey1 trace ; the lower frame shows the imaginary part
of (B.11), again the LHS of which is plotted as a solid black trace, and the RHS of which is plotted as a grey
trace.
B.3. Mechanical parameter determination (md, cd, and kd)
Once the parameter φ0 is determined, the force on the diaphragm can be deduced from the electrical
current as per (2.12). The open-circuit mechanical impedance of the loudspeaker can then be known
from:
Z mo ( s ) = φ0
ic ( s )
− Z rm ( s )
ud ( s )
(B.12)
where Zrm(s) is the mechanical-equivalent acoustic radiation impedance. The function i c ( s ) u d ( s) can
be measured directly. There are various methods for estimating Zrm(s). Typically, it is estimated by
analytical models. It may also be removed, by placing the loudspeaker-under-test in a vacuum. This
is what is typically done with microspeakers, as per Figure A.5 and Figure A.7. The following
assumes this is the case, i.e. that Zrm(s) = 0. Thus the error function for the mechanical parameters md,
cd, and kd is therefore
2
χ (md , cd , kd ) =
N stop
∑ − iω m
n
d
+ cd + i kd ωn − Z mo ( f n )
2
n = N start
The partial derivatives if χ2 with respect to md, cd, and kd are thus:
1
On colour print and electronic media, this will appear as a ‘cyan’ or ‘sky-blue’ trace.
(B.13)
Experimental determination loudspeaker parameters
183
N
stop
∂χ2
=
2ω2n md − 2kd + 2ωn Im{Z mo ( f n )}
∂md n = N start
∑
(B.14)
N
stop
∂χ2
=
2cd − 2 Re{Z mo ( f n )}
∂cd n = N start
∑
(B.15)
N
stop
k
1
∂χ2
=
− 2md + 2 d2 − 2 Im{Z mo ( f n )}
∂kd n = N start
ωn
ωn
∑
(B.16)
Re-writing these into matrix form gives
Σω2n

 0
− N

0
N
0
− N  md  − Σωn Im{Z mo ( f n )}



0   cd  =  Σ Re{Z mo ( f n )} 
Σ 1 ω2   kd   Σ 1 ωn Im{Z mo ( f n )} 
n
(B.17)
The validity of the determined parameters may be verified by comparing
?
smd + cd + kd s = Z mo ( s )
where Zmo(s) is derived from the measured FRF’s as per (B.12).
(B.18)
Appendix B
184
N/(m/s), in−phase
0.02
0.01
0
−0.01
N/(m/s), in−quadrature
−0.02
400
600
800
1000
1200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
2000
1400
1600
1800
2000
0.5
0
−0.5
Figure B.3: Open-circuit mechanical impedance estimation; upper frame: real part; lower frame: imaginary
part. In both the upper and the lower frames, the solid black trace represents the parameter fit (the LHS of
(B.18)), and the grey1 trace the measured estimate (the RHS of (B.18) ). Large frequency dependence in the
measured estimate is due to phase errors in the measurement, and additional modes in the system.
Complete assessment of the accuracy of fit also requires comparison also of the multiplicative inverse
of these quantities, i.e. the mobility. An example of such an assessment is shown in Figure B.4.
1
On colour print and electronic media, this will appear as a ‘cyan’ or ‘sky-blue’ trace.
(m/s)/N
Experimental determination loudspeaker parameters
185
1
10
200
300
400
500
600
800
1000
1500
200
300
400
500
600
Frequency (Hz)
800
1000
1500
Phase (Radians)
3
2
1
0
−1
−2
−3
Figure B.4: Mechanical mobility (open-circuit); upper frame: magnitude (log-log plot); lower frame: phase
(log-frequency plot). In both the upper and the lower frames, the solid black trace represents the parameter
fit, and the grey1 trace the measured estimate.
1
On colour print and electronic media, this will appear as a ‘cyan’ or ‘sky-blue’ trace.
Appendix B
186
(m/s)/N, in−phase
80
60
40
20
(m/s)/N, in−quadrature
0
400
600
800
1000
1200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
2000
1400
1600
1800
2000
20
0
−20
−40
Figure B.5: Mechanical mobility (open-circuit); real part (upper) and imaginary part (lower). In both the upper
and the lower frames, the solid black trace represents the parameter fit and the grey trace the measured
estimate.
B.4. References
Beranek, Leo L., Acoustics. The Acoustical Society of America, Woodbury, New York, USA (1954,
1993 ed.)
Moreno, Jorge N., “Measurement of Loudspeaker Parameters Using a Laser Velocity Transducer and
Two-Channel FFT Analysis,” Journal of the Audio Eng. Soc, Vol. 39, No. 4, pp. 243-249 (Apr.
1991)
Struck, C. J., “Determination of the Thiele-Small Parameters Using Two-Channel FFT Analysis,”
presented at the 82nd Convention of the Audio Eng. Soc., Journal of the Audio Eng. Soc.,
(Abstracts), Vol. 35 p. 386, preprint no. 2446 (May 1987)
Thiele, A. Neville, “Loudspeakers in Vented Boxes,” Proceedings of the IRE Australia, Vol. 22,
pp.487-508 (Aug. 1961). Reprinted in Journal of the Audio Eng. Soc, Vol. 19, pp. 382-392. (May
1971)
187
Appendix C. Modal analysis of loudspeaker diaphragms
C.1. Introduction
The models of loudspeakers presented in chapter 2 consider only single-degree-of-freedom dynamics
of the loudspeaker diaphragm. This assumes the diaphragm vibrates as a single rigid structure - a
valid assumption up to a certain frequency. Above some frequency, the diaphragm will vibrate in a
more complex manner. Just above this frequency up to which the diaphragm may be treated as a
single rigid structure, the vibration may be efficiently modelled by a collection of vibration modes.
These are commonly referred to as ‘break-up’ modes of a loudspeaker diaphragm. These modes may
be understood as standing waves in the diaphragm structure. Assuming an equation of motion for the
diaphragm’s structure can be defined, these modes will be given by the eigenvalues and functions of
that equation of motion. If the structure is only considered at discrete points, as is the case in FEM
modelling and experimental modal analysis, the modes will be eigenvectors. Each eigenvalue and
eigenvector will add one pole-pair and possibly one zero-pair to an s-domain representation of any
vibration frequency response function. Experimental modal analysis is the process of determining
these pole- and zero-pairs from measured frequency response functions, and subsequently deducing
properties of an appropriate equation of motion based upon the eigenvalues and eigenfunctions
defined by these pole- and zero-pairs. Experimental modal analysis may be used in this way to study
loudspeaker rocking and ‘break-up’ modes.
There are several practical problems in measuring suitable frequency response functions (FRF’s) on
loudspeakers for experimental modal analysis. Standard instruments and techniques for measurement
of these FRF’s have existed for several decades, as described by Serridge and Licht (1987), Ewins
(1988), and Bendat and Piersol (1993). These standard methods have been developed for the study of
large structures, such as automotive and aerospace frames, buildings, and bridges. Two problems are
incurred in trying to scale these standard methods down to the comparatively small loudspeaker
diaphragm: measurement of the vibration response without excessive mass-loading, and excitation of
the diaphragm by a point-force.
The first of these problems has been solved by the recently available scanning laser Doppler
vibrometer (Polytec GmbH, 2002). The standard instrument for vibration measurement – the inertial
piezoelectric accelerometer – was too large and heavy for use on loudspeaker diaphragms. This
instrument permits point-vibration-velocity measurement on loudspeaker diaphragms with sufficiently
fine spatial resolution, and without mass-loading the diaphragm.
Vibration excitation of the loudspeaker diaphragm has also remained a problem. The process for
determination of eigenvectors (mode shapes) from measured response functions assumes the vibration
response measurements are referenced to a point excitation force. On large structures, this can be
achieved with traditional shaker and referencing the response to a signal from a force transducer
connected between the shaker and the structure. Such equipment is suitable only for relatively large
loudspeaker diaphragms. For small diaphragms, there is no practical means for attaching the force
transducer to the diaphragm. In other cases, bending moments are likely to corrupt the signal
generated by the force transducer due to its large mass relative to the diaphragm.
The obvious alternative is to use the loudspeaker’s own motor structure to excite vibration in the
diaphragm. The problem with this is that it does not provide a point-force excitation. This
complicates the process of determining the loudspeaker diaphragm’s eigenvectors from the measured
response functions. Using the loudspeaker’s own motor structure requires the excitation force to be
Appendix C
188
considered as a distributed force. Details of this consideration and how it affects the process for
determining the diaphragm’s eigenvectors are presented in this appendix.
Experimental modal analysis of loudspeaker diaphragms has been studied by other authors. Bank and
Hathaway (1981) describe an early laser-based measurement system which avoids the mass-loading
problem of accelerometers, mentioned above. Bank and Hathaway used his system to measure
operational deflection shapes of a typical electrodynamic loudspeaker diaphragm. They did not
consider their measurement results in the context of experimental modal analysis. This was
considered to some extent by Struck (1990). Struck to use a laser-measurement system that providing
a direct dynamic analogue signal proportional to velocity, suitable for frequency-response
measurement with an FFT analyser. Struck also seems to be the first to subsequently perform curvefitting on the this complex frequency-response data, and showed how perturbation analysis was
possible by modifying the resulting modal mass and stiffness values. However, the above mentioned
problem of a non-point force excitation was not addressed. Skrodzka and Sek (1998) mention this
problem in their study of the vibration behaviour of a three-way loudspeaker. In their analysis, the
centre point of the low-frequency driver is taken as the point force input. Skrodzka and Sek correctly
note that the error caused by the single-point excitation can be estimated by making a [coherent] multipoint excitation in the simulation. This was not done, however, as the damping factor was the only
modal parameter of interest to the authors, which can be obtained without knowledge of the driving
point (i.e. the geometric nature of the excitation force). The vibration shapes shown by Skrodzka and
Sek are in fact the operational deflection shapes, as referenced to the input voltage, and not the modal
shapes of the mechanical systems in the loudspeaker. A generally similar method was used by
Døssing et al. (1989) to analyse the vibration characteristics of a loudspeaker enclosure.
C.2. Modal analysis of structural vibration
An equation of motion for a structure considered at M discrete-points can be written as
[ s M + sC + K ] x(s) = f (s)
2
(C.1)
where the terms in (C.1) are as follows:
M ∈ ℜ M ×M
C ∈ ℜ M ×M
K ∈ ℜ M ×M
x( s ) : C/ → C/ M
f ( s ) : C/ → C/ M
Mass matrix
Damping (viscous) matrix
Stiffness matrix
Displacement response vector – note each element of this vector is a function of the
Laplace variable s
Forcing vector– note each element of this vector is a function of the Laplace
variable s
Note that if a continuous-space model of the structure is considered, then a continuous displacement
r
response function ξ( x R , s ) is used instead of the vector x(s), and the mass, damping, and stiffness
matrices become differential operators. The forcing vector must, in this case, also be considered as a
continuously distributed forcing function. In all discussions below, it is assumed that the structure is
only considered at discrete-points, and thus the displacement response and forcing term are considered
as vectors.
Modal analysis of loudspeaker diaphragms
189
Generally, the problem to be solved is the prediction of the displacement response vector x(s) given a
specification of the forcing vector f(s). The solution can be considered as H(s), a set of M × M transfer
functions, where the jth, kth such transfer function defines the displacement response at point-j due to a
point-force at point-k, i.e.
H jk ( s ) =
x j (s)
(C.2)
f k (s)
where xj(s) is the jth element of x(s), and fk(s) is the kth element of f(s). These transfer functions can be
determined from (C.1) according to
x( s ) =
[ s M + sC + K ]
2
−1
f (s)
(C.3)
Determination of the transfer function matrix as per (C.3) is complicated by the fact that there is no
general solution for the inverse of a matrix larger than 4 ×4. Analysis of models with larger than four
degrees of freedom will, therefore require numerical evaluation at a specific frequency (value of s),
and re-inversion of this matrix for each such frequency. This problem can be overcome by
determining eigensolutions to this matrix. The process for doing this is illustrated here for the case
where the viscous-damping matrix C may be ignored. With the damping matrix C removed, the
homogeneous form of (C.1) may be expressed as this Mth-order eigenvalue problem:
[M
−1
K
] x( s ) = − s x ( s )
2
(C.4)
This eigenvalue problem, given certain conditions, will have M different solutions, each comprising an
eigenvalue ω 2m and an eigenvector φ m which are referred to as the mth solution, mth mode, or mth
eigenvalue and eigenvector. These are solutions to (C.4) as so
[M
−1
K
]φ
m
= ω m2 φ m
(C.5)
As per standard modal analysis theory (Ewins, 1988), the transfer function Hjk(s) may be determined
from the eigensolutions of (C.5) according to
H jk ( s ) =
M
φ m, j φ m,k
∑ω
m =1
2
m
− s2
(C.6)
where φ m, j is the jth element of the mth eigenvector, i.e. the jth element of φ m .
In simulation modal analysis, the eigensolutions ( ω 2m and φ m ) are determined from appropriate
descriptions of the mass and stiffness matrices, M and K. In experimental modal analysis, the
eigensolutions are determined by curve-fitting measured frequency response functions (FRF), i.e.
measurements of the FRF of the transfer function Hjk(s) in (C.2) and (C.6). The curve-fitting process
for determining these modal values from measured FRF’s is described in the next section.
C.3. Experimental modal analysis
As discussed above, in experimental modal analysis, the eigenvalues and eigenvectors are determined
from measured frequency response functions (FRF’s). A set of parameters are determined from the
FRF’s by a curve-fitting process. Details of various possible curve-fitting techniques are well
described by Ewins (1988). Given certain conditions, the original eigenvalues and eigenvectors can be
determined from the parameters determined from the measured FRF’s. These parameters are:
Appendix C
190
α jk ,m
ω 2m
ζm
‘Modal constant’
Eigenvalue, or ‘undamped natural frequency’
Damping ratio1
These parameters are used to obtain estimates of the measured FRF’s according to:
Hˆ jk ( s ) =
M
α jk , m
∑ω
m =1
2
m
− s2
(C.7)
The parameters are chosen so as to minimise the error between the measured FRF’s and the estimate
calculated according to the RHS of (C.7). Notice that in the estimate of the FRF from experimentally
determined parameters in (C.7) differs from that determined directly from the eigenvectors in (C.6).
Specifically, the numerator in the fraction on the RHS is determined by the product between two
elements of the eigenvector in (C.6), and by the modal constant α jk ,m in (C.7). In order to work out
elements of the eigenvector from the results from the curve-fitting process on the measured FRF’s it is
necessary to have measured the response at the same point at which the force was applied. If this has
been done, then one may interpret the modal constant for all modes this FRF as follows:
α kk ,m = φ 2k , m
(C.8)
With this parameter available, it is possible to calculate the elements of the eigenvector from the
modal constant according to
φ j ,m =
α jk ,m
α kk ,m
(C.9)
C.4. Interpretation of modal analysis on loudspeakers
As mentioned in the introduction to this appendix, the only practical method for vibration excitation in
small loudspeakers is to use the loudspeaker’s own motor structure. This changes the interpretation of
the results from the curve-fit parameters and the process for working out the original the eigenvectors.
Details and theoretical aspects of measuring FRF’ on a loudspeaker diaphragm are presented first.
C.4.1. Measuring structural FRF’s on loudspeakers
By using the loudspeaker’s own motor structure to excite the diaphragm, a distributed force, instead of
a point force, is applied to the diaphragm. It is generally not possible to measure this distributed force
directly, as there is no practical method of inserting a force transducer between the voice coil and the
diaphragm. Instead, the force must be indirectly deduced by measurement of the voice coil current.
As per (2.12), the force on the voice coil is given by the electrical current according to
f c ( s ) = φ 0 ic (t )
(C.10)
The voltage drop across a shunt resistor can be used to measure the electrical current, as per
Figure A.1.
The vibration response of the loudspeaker diaphragm can be measured with a scanning laser Doppler
vibrometer, as shown in Figure A.5 and Figure A.6, presented in §A.3. As described in that section,
this instrument provides a signal proportional to the velocity of the diaphragm.
1
The damping ratio ζm assumes a viscous damping mechanism in the model. This was not considered in the
theoretical presentation of eigensolutions in § C.2.
Modal analysis of loudspeaker diaphragms
191
Receptance (displacement/force) FRF’s are needed for determination of the modal parameters as per
(C.7). These may be calculated from the FRF’s measured with the system shown in Figure A.5
according to
H jc ( s ) =
Y ji ( s )
(C.11)
s φ0
where:
Hjc(s)
Yji(s)
φ0
s
receptance FRF; displacement at point j due to force from voice-coil.
FRF of velocity at point j due to voice-coil current ic(s).
transduction coefficient of loudspeaker motor (B·l product, ‘force factor’)
Laplace variable
The reasoning for the choice of notation Hjc for the receptance FRF is explained below.
C.4.2. Interpretation of curve-fit parameters
The distributed force presented by the voice-coil requires measured FRFs to be interpreted differently
from how they are in traditional modal analysis.
Measured structural FRFs are usually noted as Hjk where j indicates the point at which the response has
been measured, and k indicates the point at which the structure was excited. When the loudspeaker
diaphragm is excited by the loudspeaker’s own motor structure, the response must be considered as
that not due to force at a single point k, but a force distributed over a set of points. This set of points is
the circle forming the junction between the voice-coil and the diaphragm. These points are given the
notation c, a vector with Nc elements, and defined described as ‘the set of points on the coil,’ as so:
[
c =& p1 , p 2 ,..., p N c
]
(C.12)
where pn is the nth point on the coil. The FRFs measured on the loudspeaker diaphragm are therefore
given the notation:
H jc ( s ) =
Nc
x j (s)
∑f
n =1
pn
(C.13)
( s)
where f pn (s ) is the point-force at the point pn. According to (C.10), this force can be determined from
the measured voice-coil current according to:
f pn ( s ) =
φ0
ic ( s)
Nc
(C.14)
The modal parameters extracted from the measured receptance functions Hjc(ω) have a different
interpretation than in traditional modal analysis. The notation convention for the modal constant αjk,m
uses the first sub-script to indicate the response point (j) and the second sub-script to indicate the
excitation point (k). Thus for the loudspeaker case we write the modal constants as αjc,m., where the
second sub-script c indicates an average over the set of coil points given in (C.12). The natural
frequency, ωm, and the damping ratio, ζm, are not affected by the circular distribution of force, and
therefore in this case have the same interpretation as traditional modal analysis. Thus the estimated
FRFs are given by
Hˆ jc ( s ) =
α jc , m
M
∑ω
m =1
2
m
− ω 2 + 2iζ m ω m ω
(C.15)
Appendix C
192
As the FRFs are derived from the product between the eigenvector (or eigenfunction for continuous
systems) evaluated at the excitation point and the response point, according to the discussion above, it
is necessary to create an average of the eigenvector over the coil points. Such an eigenvector is
defined in a manner similar to (C.13),
ψ c, m
φc, m =
=
mm
1
Nc
Nc
∑φ
n =1
pn , m
(C.16)
The mode vector φc,m is simply the average of the mode shape vector for mode m at the coil points,
defined in set c in (C.12). This is the coil-force contribution factor for mode number m.
With the definition of φc,m in (C.16), the modal interpretation of the measured functions Hjc(ω) is
H jc =
φ j , m φc , m
M
∑ω
2
m
m =1
− ω2 + 2iζ mωmω
(C.17)
The driving point measurement for a loudspeaker is affected by the distributed excitation force in a
manner similar to the modal parameter extraction. A driving point measurement is that for which the
response is measured at the same point as the excitation. This enables recovery of the structures mode
shape functions from the curve-fit parameters. For the loudspeaker case, given that the force is
distributed, a distributed response is evaluated for the driving point measurement.
Summing the modal parameters at the coil points c defined in (C.12) and dividing by the number of
number of coil points provides:
α cc, m
Nc
1
=
Nc
∑α
n =1
(C.18)
pn c, m
According to the modal model, the eigenvectors (mass- and unity-normalised) are related to the modal
constant defined in (C.18) by:
α cc,m =
1
Nc
Nc
∑φ
n =1
pn , m φ c , m
= φ c2,m
(C.19)
(C.19) permits determination of the system’s eigenvectors according to both normalisation schemes.
With determination of the unity-normalised eigenvectors and adoption of a well-defined unity
normalisation convention, specification of generalised modal mass, damping and stiffness is possible.
The mass-normalised eigenvectors φn,m are determined by
φn , m =
α nc , m
(C.20)
αcc, m
Determination of the unity-normalised eigenvectors ψn,m requires adoption of a convention regarding
what ‘unity’ is. The mean-axial coil motion is adopted in this work. This normalisation has been
adopted for these two reasons:
•
Compatibility with lumped-parameter modelling conventions.
•
Simplicity in coupling to the electrical domain.
The mean-axial coil motion unity normalisation convention requires that the unity-normalised
eigenvectors ψn,m be unity when averaged over the set of points on the coil, c. This is defined by:
ψc, m
1
=
Nc
Nc
∑ψ
n =1
pn , m
=1
(C.21)
Modal analysis of loudspeaker diaphragms
193
With this definition, the modal mass may be determined from the drive-point modal parameter αcc
according to (C.19).
mm =
1
αcc, m
(C.22)
With the modal-mass definition in (C.22), the unity-normalised eigenvectors can be determined from
the modal parameters according to
ψ n,m =
α nc, m
(C.23)
αcc, m
C.4.3. Differential vs. single-ended considerations
A lumped parameter mechanical model which is equivalent to the modal model , when driven by the
voice-coil, can be derived. The lumped parameter modal treats the voice-coil as a lumped mass. With
such a definition, it becomes identical to the drive point measurement, which is given by the
generalised modal mass, damping, and stiffness:
M
H cc =
∑s
m =1
1
2
m m + sc m + k m
(C.24)
The modal mass mm is given by (C.22). The modal damping cm is given by
c m = ζ m ω m α cc,m
(C.25)
The modal stiffness km is given by
2
k m = ωm
α cc,m
(C.26)
An electrical circuit can be constructed which is equivalent to the modal model of the diaphragm. An
example showing an electrical analogy of a modal model with three modes is shown in Figure C.1.
This analogous electrical circuit illustrates how the mechanical modes couple to the electrical and
acoustical domains.
Figure C.1: Equivalent circuit representation of a modal model of a loudspeaker diaphragm. This circuit is
equivalent to a modal model which contains three (3) modes.
Appendix C
194
For each mode in the mechanical modal model, there is an analogous current loop in the electrical
circuit. Each current loop corresponds to one mode in the modal model. For each current loop m,
there is an inductor Lm, resistor Rm, and capacitor (condenser) Cm, and an effective acoustic radiating
area Sm.
The inductor Lm simulates the behaviour of the modal mass mm, and is thus given by
L m = 1 α cc, m
(C.27)
The resistor Rm simulates the modal damping for each mode m, and is thus given by
Rm = cm = ζ m ωm αcc, m
(C.28)
The capacitor (condenser) Cm simulates the modal stiffness for each mode m, which is
C m = (k m )
−1
= α cc, m ω 2m
(C.29)
The effective acoustic radiating area Sm is given by the product of mean-value of the unity-normalised
eigenvector for mode m and the total area which is covered by the eigenvector, S0.
Sm =
S0
N
N
∑ψ
n, m
(C.30)
n =1
In (C.30), N is the total number of elements of the eigenvector ψn,m. Determination of the total area
covered by the eigenvector S0 requires analysis of the geometry corresponding to the eigenvector.
C.5. Summary
Experimental techniques for obtaining FRF’s from loudspeaker diaphragms suitable for modal analysis
are reviewed. A method for interpreting modal analysis results when the loudspeaker’s own motor
structure is used for force excitation is presented. Methods for interpreting these results in terms of
traditional lumped-parameter (equivalent electrical circuit) models are presented.
C.6. References
Bendat, Julius S., and Allan G. Piersol, Engineering Applications of Correlation and Spectral
Analysis, 2nd Ed., John Wiley & Sons., Inc., NY, NY, USA. (1993)
Bank, G., and G. T. Hathaway, “A Three Dimensional Interferometer Vibrational Mode Display,” J.
Audio Eng. Soc., 29, pp. 314-319. (1981)
Døssing, Ole, Christian Hoffman, Lars Mattiessen, and Ole Jacob Veiergang, “Measurement of
Operating Modes on a Loudspeaker Cabinet”, presented at the 87th Convention of the Audio
Engineering Society, preprint no. 2848. (18-21 Oct. 1989)
Ewins, D. J., Modal Testing: Theory and Practice, Research Studies Press Ltd., Letchworth,
Hertfordshire, England. (1988)
Hewlett Packard Inc. The Fundamentals of Modal Testing, Application Note 243-3. (1991)
Heylen, W., S. Lammens, and P. Sas (eds.), “Modal Analysis Theory and Testing,” ISBN 90-7380261-X, K. U. Leuven. (1997)
Larsson, Daniel, “Using Modal Analysis for Estimation of Anisotropic Material Constants.” Journal
of Engineering Mechanics, 123, No. 3. (March 1997)
Polytec GmbH, “Laser Doppler Vibrometer,” Polytec GmbH, Polytec-Platz 1-7, 76337 Waldbronn,
Germany. (2002)
Modal analysis of loudspeaker diaphragms
195
Serridge, Mark, and Torben R. Licht, Piezoelectric Accelerometer and Vibration Preamplifier
Handbook, Brüel & Kjær A/S (printed by K. Larsen & Søn A/S · DK 2600 Glostrup) (Nov. 1987)
Skrodzka, E. Eb., and A. P. Sek “Vibration Patters of the front panel of the loudspeaker system:
Measurement conditions and results.” Journal of the Acoustical Society of Japan, 19, pp. 249-257.
(July 1998)
Struck, Christopher J., “Investigation of the Nonrigid Behaviour of a Loudspeaker Diaphragm Using
Modal Analysis,” Journal of the Audio Eng. Soc., 8, pp. 667-675. (Sept. 1990)
196
Appendix C
197
Appendix D. Rocking modes in single-suspension loudspeakers
D.1. Introduction
This appendix discusses the problem of rocking modes in single suspension loudspeakers, develops a
theoretical model of rocking modes, and describes a method for measuring rocking modes
experimentally. Rocking modes are problematic in single-suspension loudspeakers because their
rotational vibration is not impeded by a spider, as in traditional loudspeakers. The basic construction
principle of a single-suspension loudspeaker is contrasted with a traditional electrodynamic directradiating loudspeaker in Figure D.1.
Figure D.1: Single-suspension loudspeaker vs. traditional electrodynamic direct radiator loudspeaker.
The current work is motivated by a study of loudspeakers used in hand-held mobile phones. However,
single-suspension loudspeakers are found in other incarnations of electrodynamic loudspeakers, such
as horn compression drivers and dome tweeters.
In Chapter 4, Section 7, of McLachlan's book 'Loud Speakers,' the subject of the various rigid-body
modes of vibration are considered. McLachlan studies the rigid body modes by analogy to a twodimensional mechanical system of a lumped mass supported by two membranes. Using this model,
McLachlan predicts the natural frequency of the rocking mode (‘wobble’ in MacLachlan’s text) to
occur at a frequency relative to the fundamental mode of vibration of 3 m s m d or ‘about twice’ as
written by McLachlan (1934). In the preceding section of MacLachlan’s book, the construction of
‘centring devices’ is introduced. The ‘centring device,’ nowadays referred to as a ‘spider,’ is
explained by McLachlan to ‘preserve axial motion of the coil, thereby eliminating wobble…’
It is interesting to notice the absence of a discussion on spiders (or ‘centring devices’) in other classic
texts on Loudspeakers e.g. Beranek (1954), Hunt (1954), and Olson (1957). The actual author’s
deduction from this absence is that during the 20 years between MacLachlan’s and the other classic
texts, the necessity of a spider to the operation of loudspeaker became widely accepted. To imagine a
direct radiator loudspeaker without a spider would be as unthinkable as an automobile without brakes.
One may view, therefore, the renewed popularity of loudspeakers without spiders as at worst a step
backwards in technology, and at best a venture into known dangerous waters.
D.2. Low frequency modes of vibration of a single suspension loudspeaker
As stated in the introduction, in a single-suspension loudspeaker, diaphragm rocking, or ‘wobble,’ will
occur at approximately 1.5 ~ 2 times the fundamental resonance frequency. A theoretical model of the
rocking modes can be developed using rigid body mechanics. The first step in this process is to
198
Appendix D
establish the interaction between translational and rotational modes of vibration. The second step is to
describe this interaction in terms of the physical characteristics of a loudspeaker.
D.2.1. Translational vs. rotational modes of vibration
To understand the behaviour of rocking modes, it is necessary to be familiar with the difference
between rotational and transitional motion. We may describe the in-plane motion of a lumped massspring damper system as shown in Figure D.2 with a linear, translational component of motion x(t),
and a rotational component motion θ(t).
Figure D.2: Linear (translational) and rotational in-plane modes of vibration.
From Tse et al. (1978), the equations of motion of this system in matrix form are
k1 + k2
− (L1k1 − L2 k2 )  x  0
m 0   x  
s2 
+



2
2   =   .
 0 J  θ − (L1k1 − L2 k2 ) k1L1 + k2 L2  θ 0
(D.1)
Attention here is drawn to the off-diagonal coupling terms in the stiffness matrix in (D.1), which are
both –(L1k1-L2k2). The distances L1 and L2 are the those between the mass’ centre of gravity and the
springs k1 and k2 respectively. If L1k1=L2k2, then a force applied in the +x direction will produce only a
linear, translational deflection. Conversely, if a torque, or moment, is applied at exactly the centre of
gravity, it will produce only a rotational deflection. However, if L1k1 ≠ L2k2, force applied to the centre
of gravity will produce rotational in addition to linear deflection and vice versa for a torque, or
moment, applied at the centre of gravity.
Herein lies a key to understanding the problem of rocking modes in loudspeakers. Although the motor
unit of a loudspeaker produces no torque or direct moment on the diaphragm, an asymmetric
diaphragm stiffness or an off-balance drive position will cause a moment to be imposed on mass of the
diaphragm.
It should be noted that the coupling terms between the equations of motion are characteristic of the
choice of co-ordinate system, chosen here for convenience. The diaphragm will vibrate in its own
natural manner, independent of the co-ordinate system. Note also that coupling between the equations
of motion is not equivalent to coupling between the natural modes of vibration, which is a different
property of the diaphragm. A good discussion of this technique is available in Tse et al.
D.2.2. ‘Rocking’ modes of diaphragm vibration
By extending the model developed above to two dimension, we can develop a model for the rocking of
a diaphragm coil on its suspension. We consider the linear, translational, or ‘axial’ mode of vibration
along the z-axis and two rotational modes of
vibration, one about the x-axis and the other about the y-axis. This is shown diagrammatically in
Figure D.3
Rocking modes in single-suspension loudspeakers
199
Figure D.3: Diagram of rocking modes in a single-suspension loudspeaker diaphragm.
For the speakers under investigation in this paper, the diaphragm is constructed of very thin plastic.
This makes the diaphragm mass much smaller than that of the coil. Therefore only the inertial
component of the coil mass is considered in the mechanical dynamics. The effective mass moment of
inertia of the coil about the x- and y-axes is assumed to be that of a hollow cylinder
(
1
J = 12
m 3r12 + 3r22 + h 2
)
(D.2)
This assumption does not limit other aspects of the current study of single-suspension loudspeakers, as
they are valid for other distributions of the mass moment of inertia.
Ideally, the two rocking modes about θx and θy are degenerate. That is, they occur at the same
frequency, and their mode-shapes are identical, except for a right-angle difference between them. In
experimental modal analysis, they will appear as one mode; it is not possible to distinguish between
them unless the structure is excited in more than one location. This is generally not possible to do on
small loudspeaker diaphragms. In some cases, however, if the material stiffness is not completely
isotropic, these normally degenerate modes will ‘split,’ i.e. occur at different frequencies. This
property can be used to determine anisotropy the membrane stiffness (Larsson, 1997).
In general, rocking modes are more problematic in telecom-type speakers than most incarnations of
single-suspension speakers for consumer or commercial music reproduction or public address such as
dome-tweeters. This is due to the use of ferro-fluids in the air gap, which provide a high degree of
damping for the rocking modes.
We consider the suspension between the coil and mounting edge to have a stiffness distribution
dependent upon the angle around the coil, α. This dependence of the suspension on this angle is
represented as a stiffness distribution function σ(α) . Any stiffness distribution around the coil can be
represented by a Fourier series of sinusoidal functions around the angle α:
σ( α ) =
∞
∑σ
n
cos(nα + φn )
(D.3)
n =0
It will be shown that the dominant features of rocking modes can be described with only the first three
terms of the series in (D.3), so that
σ(α) = σ0 + σ1 cos(α + φ1 ) + σ 2 cos(2α + φ2 )
(D.4)
For example, the stiffness distributions for σ0=500N/m2, σ1=σ2=150N/m2, φ1=φ2=0 are shown in
Figure D.4. The dimension of the distribution functions are N/m2, for stiffness per unit length along
the diameter path.
Appendix D
200
Figure D.4: Stiffness distributions for given values of σ.
The equations of motion for the three degrees of freedom may be represented in matrix form as
2
s Mx + Kx = f
⇒
m
0
s 0
0

Jx
2
0
0  z   k z
  
0   θ x  +  k zθ x
J y  θ y   k zθ y

k zθ x
k θx
k θ xθ y
k zθ y   z  φ 0 
 
k θ x θ y   θ x  =  0 
k θ y  θ y   0 

(D.5)
where the notation has the following definitions:
s
The Laplace variable. s= iω, i = − 1 , ω = 2πf, f = frequency in Hz.
φ0
Transduction coefficient.
m
The lumped mass of the diaphragm, excluding the effects of any air loading.
Jx
The mass moment of inertia of the diaphragm about the x-axis.
Jy
The mass moment of inertia of the diaphragm about the y-axis.
Stiffness of the membrane (= 1/compliance), excluding any ‘air cushion’ on which the
kz
loudspeaker is mounted,
opposing linear (axial) motion of the diaphragm.
k θ x Rotational stiffness of the membrane about the x-axis.
kθy
Rotational stiffness of the membrane about the y-axis.
k zθ x Coupling stiffness between translational motion along the z-axis, and rotational motion about
the x-axis.
k zθ y Coupling stiffness between translational motion along the z-axis, and rotational motion about
the y-axis.
k θ xθ y Coupling stiffness between rotational motion about the x-axis and rotational motion about the yaxis.
Rocking modes in single-suspension loudspeakers
201
All of the terms in the stiffness matrix in (D.5) and described above can be derived from the stiffness
distribution function in (D.4). The terms are derived using the same strategy as the stiffness terms for
the simple mass-and-spring system described in (D.1). However, because the stiffness is
continuously distributed along the diaphragm’s edge, the stiffness terms must be determined by
integrals instead of sums.
The first stiffness term to be defined is that opposing linear (axial) motion of the diaphragm. This
term is equal to the inverse of the mechanical compliance of the diaphragm:
2π
∫
k z = σ(α)dα = 2πσ0 = 1 CMD
(D.6)
0
Rotational stiffness is in general defined as the integral of the product between a stiffness distribution
and the square of a distance function, giving the distance between the stiffness and the axis of rotation.
∫
k θ = (axis to stiffness distance) 2 (stiffness distribution )dx
x
(D.7)
For the loudspeaker, we want to integrate around the path of the diaphragm edge, where the stiffness is
defined by the stiffness distribution function in (D.4). The distance between the x-axis and the
stiffness may be defined in terms of the angle α, as shown in Figure D.4, according to r·sin(α).
Therefore the rotational stiffness is defined as
2π
∫
kθ x = r 2 sin 2 α σ(α) dα = πr 2σ0 +
0
π 2
r σ 2 cos φ2
2
(D.8)
The same technique can be applied to determine the rotational stiffness about the y-axis
2π
∫
kθ y = r 2 cos 2 α σ(α) dα = πr 2σ0 −
0
π 2
r σ 2 cos φ2
2
(D.9)
The coupling terms are defined in the same manner as a (D.1), except again using integrals.
2π
∫
k zθ x = −r sin ασ(α)dα = πrσ1 sin φ1
(D.10)
0
2π
∫
k zθ y = −r cos ασ(α) dα = πrσ1 cos φ1
(D.11)
0
The coupling term between the two rotational modes is similarly defined:
2π
kθ x θ y = − r
2
π
∫ sin α cos ασ(α)dα = − 2 r σ
2
2 sin φ 2
(D.12)
0
From the foregoing analysis, the impact of the terms in the stiffness distribution of (D.4) can be
qualitatively described.
σ0
Fundamental resonance frequencies
σ1
Coupling between translational (axial) and rotational modes.
σ2
modes
Splitting of frequencies of x-axis and y-axis rotational modes; coupling between rotational
Appendix D
202
D.3. Eigenvalue analysis
Traditional eigenvalue analysis of equations in (D.5) is straightforward. This system has three
eigenvalues, ω2m (for m = 1 to 3), and a 3 × 3 eigenvector matrix Ψ, for which the mth column of Ψ
corresponds to eigenvalue ω2m .
If the off-diagonal terms of K in (D.5) are zero, there will be no coupling between translational and
rotational degrees of freedom. This will be reflected in Ψ; which will also be diagonal. If offdiagonal terms are introduced to K,
We may look at Ψ by analysing linear motion on the +z direction caused by small angular oscillations.
Thus we analyse modified eigenvectors
1 0 0 

0  Ψ
lin Ψ = 0 r2
0 0 r2 
D.4. Problems with rocking modes
D.4.1. Mechanical tolerances
The magnitude of coil rocking can be sufficiently large to cause contact between the coil and/or
diaphragm and the magnet assembly, as shown in Figure D.5. Contact between the coil and/or
diaphragm and the magnet assembly will produce unacceptably high distortion. It may also cause
damage to the coil or diaphragm, which can result in failure of the loudspeaker. This may necessitate
a widening of the coil gap, which is significantly disadvantageous, as this will reduce the magnetic
field strength in the coil gap.
Figure D.5: Deflection in severe rocking mode may cause contact between coil and magnet assembly.
D.4.2. Acoustic radiation
A rocking mode can create acoustic radiation either by monopole or dipole radiation. Monopole
radiation is only possible when the shape of natural rocking mode is not completely symmetric. This
tends to occur when there is a large cos(α) asymmetry in the suspension stiffness distribution.
Rocking modes in single-suspension loudspeakers
203
D.5. References
Beranek, Leo L., Acoustics, The Acoustical Society of America (1954).
Ewins, D. J., Modal Testing: Theory and Practice Research Studies Press Ltd., Letchworth,
Hertfordshire, England (1988).
Hunt, Frederick V., Electroacoustics: The Analysis of Transduction, and Its Historical Background.
Harvard University Press, Cambridge, Mass., USA. (1954)
Larsson, Daniel, “Using Modal Analysis for Estimation of Anisotropic Material Constants.” Journal
of Engineering Mechanics, 123 (3). (March 1997)
McLachlan, N. W., Loud Speakers, Theory, Performance, Testing, and Design. pp. 69 – 72. Oxford
University Press, London, U. K. (1934).
Olson, Harry F., Acoustical Engineering. D. Van Nostrand Company, Inc. (1957)
Tse, Francis S. et al. Mechanical Vibrations p. 156. Prentice-Hall Inc., Englewood Cliffs, New Jersey
(1978).
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