Electroacoustic absorbers: bridging the gap between shunt loudspeakers and active sound absorption.

Abstract

The acoustic impedance at the diaphragm of an electroacoustic transducer can be varied using a range of basic electrical control strategies, amongst which are electrical shunt circuits. These passive shunt techniques are compared to active acoustic feedback techniques for controlling the acoustic impedance of an electroacoustic transducer. The formulation of feedback-based acoustic impedance control reveals formal analogies with shunt strategies, and highlights an original method for synthesizing electric networks ("shunts") with positive or negative components, bridging the gap between passive and active acoustic impedance control. This paper describes the theory unifying all these passive and active acoustic impedance control strategies, introducing the concept of electroacoustic absorbers. The equivalence between shunts and active control is first formalized through the introduction of a one-degree-of-freedom acoustic resonator accounting for both electric shunts and acoustic feedbacks. Conversely, electric networks mimicking the performances of active feedback techniques are introduced, identifying shunts with active impedance control. Simulated acoustic performances are presented, with an emphasis on formal analogies between the different control techniques. Examples of electric shunts are proposed for active sound absorption. Experimental assessments are then presented, and the paper concludes with a general discussion on the concept and potential improvements.

Full text

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1 Electroacoustic absorbers: Bridging the gap between shunt
2 loudspeakers and active sound absorption
3
Herve ´ Lissek,a)Romain Boulandet, and Romain Fleury
Laboratoire d’Electromagne ´tisme et d’Acoustique, Ecole Polytechnique Fe ´de ´rale de Lausanne, Station 11,
CH-1015 Lausanne, Switzerland
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(Received 10 June 2010; revised 18 February 2011; accepted 24 February 2011)
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The acoustic impedance at the diaphragm of an electroacoustic transducer can be varied using a
range of basic electrical control strategies, amongst which are electrical shunt circuits. These
passive shunt techniques are compared to active acoustic feedback techniques for controlling the
acoustic impedance of an electroacoustic transducer. The formulation of feedback-based acoustic
impedance control reveals formal analogies with shunt strategies, and highlights an original method
for synthesizing electric networks (“shunts”) with positive or negative components, bridging the
gap between passive and active acoustic impedance control. This paper describes the theory unify-
ing all these passive and active acoustic impedance control strategies, introducing the concept of
electroacoustic absorbers. The equivalence between shunts and active control is first formalized
through the introduction of a one-degree-of-freedom acoustic resonator accounting for both electric
shunts and acoustic feedbacks. Conversely, electric networks mimicking the performances of active
feedback techniques are introduced, identifying shunts with active impedance control. Simulated
acoustic performances are presented, with an emphasis on formal analogies between the different
control techniques. Examples of electric shunts are proposed for active sound absorption. Experi-
mental assessments are then presented, and the paper concludes with a general discussion on the
concept and potential improvements.
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C 2011 Acoustical Society of America. [DOI: 10.1121/1.3569707]
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PACS number(s): 43.38.Ew, 43.50.Ki, 43.20.El [AJS]Pages: 1–11
26 I. INTRODUCTION
27
In the pioneering active noise control patent,1the under-
28 lying principle lies in the processing of acoustic interfer-
29 ences, i.e. the cancellation of a primary sound wave with a
30 controlled secondary source. This early formulation of active
31 noise cancellation paved the way to a broad variety of active
32 concepts, among which is active sound absorption. The for-
33 mal introduction of the concept of sound absorption by elec-
34 troacoustic means can be attributed to Olson and May,2who
35 applied a feedback control on a loudspeaker, based on sound
36 pressure sensing. In the footsteps of this novel formulation
37 of active noise control, Jessel et al.3studied the principle of
38 active absorbers in the light of formal analogies with the
39 Huygens theory of sound propagation, leading to practical
40 requirements of the secondary sound sources. Guicking4
41 extended the concept to an hybrid structure combining an
42 acoustic passive absorber with an active electroacoustic
43 transducer. This principle slightly evolved in the 1980s to
44 turn into the concept of “smart foam.”5This concept aims at
45 modifying the acoustic behavior of a polymer structure
46 through a thin embedded polyvinylidene fluoride (PVDF)
47 film and an acoustic feedback, where the whole acts both as
48 a passive device and an active sound absorber.
At the same time, feedback techniques were developed
50 introducing the concept of “direct impedance control” on a
51 loudspeaker diaphragm.6Here, the feedback employs a com-
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52
bination of sound pressure and diaphragm velocity sensing,
resulting in a broadband acoustic impedance control.7–9In
these feedback techniques, the collocation of the actuator
and the sensor plays an important role in the stability and in
the performances of the controlled device, especially in the
cases where the active elements are distributed within
arrays.10This has inspired advanced control techniques,
where the actuator is capable of self-sensing acoustic quanti-
ties. Indeed, the actuator is among one of the numerous com-
ponents that rule the performances of active noise control,11
but the active noise control algorithms generally take little
account of its dynamics and the means to modify its passive
response to external sound pressure. Therefore it seems im-
portant to develop versatile techniques for better control of
the loudspeaker dynamics that can be referred to as
“actuator-based” active impedance control.
The first realizations of an actuator-based active feed-
back control can be found in the realm of audio engineer-
ing.12,13In Ref. 13, an original velocity-feedback technique
was employed with the objective of further extending the
response of a loudspeaker in the low-frequency range. In this
setup, the velocity information was processed through the
differential voltage of a Wheatston bridge at the electrical
terminals of the transducer, based on assumptions concern-
ing the loudspeaker dynamics (i.e., motional feedback). This
feedback resulted in a damping of the velocity response of
the actuator around the resonance frequency, thus reducing
the non-linear behavior in the low-frequency range. These
actuator-based concepts also provide a smart and efficient
solution for sensing acoustic quantities out of an electrical
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a)Author to whom correspondence should be addressed. Electronic mail:
herve.lissek@epfl.ch.
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82 filter. For example, Lane et al.14use a physical model of the
83 electroacoustic loudspeaker within an active noise control
84 device, with the objective of designing specific electrical fil-
85 ters dedicated to extract acoustic information out of electri-
86 cal sensing.
In 1970, Bobber15showed that a transducer shunted by
88 an electronic generator could be used for matching the trans-
89 ducer acoustic impedance to the sound field. Instead of tar-
90 geting a local sound pressure reduction (as in Olson and
91 May’s paper2), the active transducer is used to modify the
92 acoustic particle velocity through a passive absorber to avoid
93 any perturbation of the incident sound wave. When the con-
94 ditions for impedance match are met, the electronic system
95 operates as the characteristic impedance of an acoustic trans-
96 mission line, thus serving as an absorber of sound energy. In
97 the footsteps of this work, Elliott et al.16suggested that a
98 secondary sound source could be seen as an electrical net-
99 work designed in order to match a specific load that maxi-
100 mizes the absorbed sound power. These observations paved
101 the way to straightforward strategies for sound absorption
102 through electroacoustic means, such as the concept of “shunt
103 loudspeakers.”17It is proven that a simple electric resistance
104 of positive value connected to the electric terminals of a
105 loudspeaker can modify the value of the acoustic impedance
106 of the diaphragm up to the point at which the loudspeaker
107 system becomes an excellent absorber around its resonance
108 frequency. Employing “negative” resistances (through nega-
109 tive impedance converters) further varies the acoustic im-
110 pedance of the device.15
111 loudspeakers can also be enhanced with an “hybrid
112 feedback,”18,19in which the acoustic impedance of a loud-
113 speaker is broadly modified by connecting a negative resist-
114 ance in series with a sound pressure-feedback. Since the
115 negative impedance realizes motional feedback,13the acous-
116 tic performances obtained with hybrid feedback are similar
117 to the above-mentioned direct impedance control techniques.
All these passive and active control strategies have been
119 extensively studied, but the equivalence between shunt and
120 feedback control is not straightforward and is only suggested
121 in a few papers. For example, Bobber15presents electrical
122 networks connected to a loudspeaker that allow the matching
123 of the acoustic impedance of its diaphragm to the medium.
124 However, a conceptual bridge between these different con-
125 trol techniques is sill missing, which is one of the motiva-
126 tions of the present paper.
In the following, the “electroacoustic absorber” (EA)
128 concept is introduced, inspired by Olson and May’s and
129 Bobber’s articles,2,15as well as by the techniques for substi-
130 tuting a negative resistance for a velocity-feedback presented
131 by Lissek and Meynial.18,20Section II introduces a strategy
132 for synthesizing the acoustic impedance presented by a loud-
133 speaker to the medium that can be performed by setting three
134 independent parameters of a one-degree-of-freedom resona-
135 tor. A synthetic formulation of the acoustic admittance
136 resulting either from an electric shunt or an acoustic feed-
137 back is specifically developed. Then, acoustic feedbacks are
138 shown to be equivalent to electric shunts that are formalized
139 hereafter, allowing the synthesis of electric networks in view
140 of active sound absorption. These developments are
87
The performances of shunt
118
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141
followed by computational and experimental validations, as
well as practical discussions on this unifying concept, with
an emphasis on the analogies between the different control
strategies.
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II. ELECTROACOUSTIC ABSORBER CONCEPT
146
A. General presentation
147
In this paper, we consider a closed-box moving-coil
loudspeaker radiating in a waveguide of adapted cross-sec-
tion, where the assumption of plane waves under normal
incidence is used throughout the formulations. Moreover,
the following developments are restricted to the case of an
electrodynamic transducer (see Fig. 1), but the presented
results are also transposable to other transduction cases.15,21
The discussions focus on different feedback settings at the
electric terminals of an electrodynamic transducer: A passive
resistance, a velocity-feedback, and a direct active imped-
ance control. The system as a whole (the electroacoustic
transducer, the enclosure, and the electric feedback) is
referred to as the electroacoustic absorber (EA).
A closed-box electrodynamic loudspeaker is a linear
time-invariant system that, under certain hypotheses, can be
described with differential equations.22From Newton’s law
of motion, the mechanical dynamics of the loudspeaker dia-
phragm, for small displacements and below the first modal
frequency of the diaphragm, can be modeled with the fol-
lowing linear differential equation:
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SpþðtÞ ¼ ?Mms_ tðtÞ ? RmstðtÞ
CmsþqC2S2
?
1
Vb
? ?ð
tðtÞ ? dt ? BliðtÞ;
(1)
167
where q is the density of the medium and c is the celerity of
sound in the medium, Mms, Rms, and Cmsare the mass, me-
chanical resistance, and compliance of the moving bodies of
the loudspeaker, Vbis the volume of the cabinet, v(t) is the
diaphragm velocity (opposed to total particle velocity), Bl is
the force factor of the transducer (where B is the magnetic
field magnitude and l is the length of the wire in the voice
coil), i(t) is the driving current, Bli(t) being the Laplace force
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FIG. 1. Description of the electroacoustic loudspeaker and definition of
parameters.
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175 induced by the current circulating through the coil, S is the
176 effective piston area, and pþis the sound pressure at the
177 outer (front) surface of the loudspeaker.
The electrical dynamics can also be modeled by a first-
179 order differential equation given as:
178
eðtÞ ¼ ReiðtÞ þ LediðtÞ
dt
? BlvðtÞ;
(2)
180 where e(t) is the voltage applied at the electrical terminals,
181 Reand Leare the dc resistance and the inductance of the
182 voice coil, respectively, and Blv(t) is the back electromotive
183 force (EMF) induced by its motion within the magnetic field.
Equations (1) and (2) form a system of differential equa-
185 tions describing the loudspeakers dynamics. Expressing the
186 preceding relationships with the use of Laplace transform
187 yields the characteristic equations of the electrodynamic
188 loudspeaker, given as:
184
SPþðsÞ ¼ ? sMmsþ Rmsþ
EðsÞ ¼ ðsLeþ ReÞIðsÞ ? BlVðsÞ
1
sCms
??
VðsÞ ? BlIðsÞ
(
;
(3)
189 where Pþ(s), V(s), E(s), and I(s) are the Laplace transforms
190 of
pþ(t),
v(t),
e(t),
191 1=ðCmcÞ ¼ 1=ðCmsÞ þ qc2S2=ðVbÞ is the equivalent compliance
192 due to the closed-box at the rear side of the loudspeaker.
The analytical formulation of the loudspeaker system
194 can then be illustrated in the form of an equivalent circuit
195 illustrated in Fig. 2, where esand Rsare the voltage source
196 and its internal resistance, respectively. On the acoustic
197 side, we assume an ideal exogenous sound source, pS,
198 located at one extremity of a waveguide facing the EA.
199 The total acoustic pressure at the front side of the loud-
200 speaker diaphragm, pþ, therefore corresponds to the addi-
201 tion of an incident sound pressure piand a reflected sound
202 pressure pr, which accounts for the mechanical radiation
203 impedance of the front face of the loudspeaker. However,
204 this radiation impedance is excluded from the studied sys-
205 tem with a view of providing general properties of the
206 sound absorber (apart from the radiation conditions of the
207 diaphragm). This impedance would not appear in the
208 following developments.
and
i(t),respectively,and
193
209
B. Formulation of electroacoustic absorbers
210
1. Acoustic absorption capability of the speaker face
211
It is always possible to derive the system of Eq. (3) in
order to write the normalized acoustic admittance of the
loudspeaker face as a function of the sound pressure Pþ(s)
and velocity V(s), whatever the load or feedback at its elec-
trical terminals,
212
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215
YðsÞ ¼ ?qc ?VðsÞ
PþðsÞ:
(4)
216
The minus sign is justified by the fact that V(s) is defined as
the diaphragm velocity, opposed to the total particle velocity
at the diaphragm. The corresponding reflection coefficient
can be derived after
217
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219
rðsÞ ¼1 ? YðsÞ
1 þ YðsÞ:
(5)
220
The extraction of the magnitude jr(f)j of r(s) yields the
sound absorption coefficient a(f)
221
aðfÞ ¼ 1 ? jrðfÞj2;
(6)
222
valid for the steady-state response of the system to harmonic
excitations.
Equations (3)–(6) indicate that the choice of the electric
load imposes certain absorption characteristics at the loud-
speaker front face. The term “electroacoustic absorber” is
thus justified when this load impedance Z(s) is tailored in
such a way as to exhibit positive values of the acoustic
absorption coefficient.
This section aims at providing the general expression of
the acoustic admittance presented at the loudspeaker dia-
phragm, when its electric terminals are connected to electric
networks or acoustic feedback voltages. We consider here
the voltage at the terminals of the loudspeaker as the combi-
nation of
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235
236237
(1)
(2)
a feedback voltage on diaphragm velocity;
a feedback voltage on sound pressure at the front face of
the diaphragm;
the source voltage lowering induced by the source elec-
tric resistance Rs.
238239
240
241242
(3)
243
244
The input voltage is therefore given by
EðsÞ ¼ EsðsÞ ? RsIðsÞ
¼ CvVðsÞ þ CpPþðsÞ ? RsIðsÞ;
(7)
245
where Cvand Cprepresent the feedback gains, respectively,
in Vm?1s and VPa?1, including sensors sensitivities.
By replacing voltage E(s) in Eq. (3) with the expression
of Eq. (7) yields the normalized acoustic admittance:
246
247
248
YðsÞ ¼ Zmc
s2a2þ sa1
s3b3þ s2b2þ sb1þ b0;
(8)
FIG. 2. Circuit representation of an electrodynamic loudspeaker including
the electric load (electric mesh at the right of the loudspeaker). The acoustic
disturbance is represented here by an ideal source of sound pressure pS, and
the waveguide is represented by a two-port system, accounting for the rela-
tionship between the input sound pressure pSand the sound pressure pþat
the front of the loudspeaker.
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249 with
a2¼ Le
a1¼ Reþ Rsþ CpBl
b3¼ LeMms
b2¼ ðReþ RsÞMmsþ LeRms
b1¼ ðReþ RsÞRmsþLe
b0¼Reþ Rs
S
Cmsþ BlðBl þ CvÞ
Cmc
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
>
<
>
>
>
>
>
>
>
;
(9)
250 where Zmc¼qcSis the mechanical equivalent to characteris-
251 tic medium impedance Zc¼qc. This expression can be sim-
252 plified in the low-frequency range, below the cut-off
253 frequencies feand fmeof the two electrical filters resulting
254 from the connection of the electro–mechanical transducer to
255 the electric load of Eq. (7) determined by,
f < fe¼1
f < fme¼1
2p
Reþ Rsþ CpBl
Le
Reþ Rs
Le
S
2p
þRms
Mms
? ?
8
>
>
>
>
:
<
;
(10)
256 and
257 fs¼ 1=ð2p
258 generally being of the order of magnitude of 6 X and induct-
259 ance Leof about 1 mH, the above-mentioned cut-off frequen-
260 cies are in the range of 1 kHz. The higher order terms in the
261 numerator and denominator of Eq. (8) can be neglected, thus
262 justifying the simplification of Y(s) as
especially
p
around
. Without feedback, dc resistance Re
the resonance frequency
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MmsCmcÞ
YðsÞ ? Zmc
s
s2MmEAþ sRmEAþ
1
CmEA
;
(11)
263 where
MmEA¼ Mms
Reþ Rs
Reþ Rsþ CpBl
S
RmEA¼ðReþ RsÞRmsþLe
?
Cmcþ BlðBl þ CvÞ
Reþ Rsþ CpBl
CpBl
SðReþ RsÞ
S
CmEA¼ Cmc 1 þ
?
8
>
>
>
>
>
>
:
>
<
>
>
>
>
>
>
>
>
>
>
>
(12)
264 are the mechanical equivalent components of the EA which
265 exhibits the characteristics of a resonator. This set of param-
266 eters can also be replaced by the following set of parameters:
fEA¼
1EA¼RmEA
1
2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffi
MmEACmEA
p
Zmc
QEA¼
1
RmEA
MmEA
CmEA
r
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
:
;
(13)
267
fEAbeing the resonance frequency, fEAbeing the normalized
268 resistance, and QEAbeing the resonance quality factor of the
269 EA.
270
Thus, the EA concept allows easy determination of the
parameters Rs, Cv, and Cp, in order to match the desired
resonator parameters (fEA, fEA, QEA), opening the way to a
straightforward control strategy for the active absorption
of sound. A very singular result is the fact that the reso-
nance frequency fEAis not affected much by the control,
where any increase of pressure-feedback gain Cpin posi-
tive values leads to the reduction of the apparent mass of
the EA together with the increase of its compliance, thus a
consecutive increase of the control bandwidth (inversely
proportional to QEA). The sensitivity of the EA performan-
ces with the three control parameters is further detailed in
Sec. III.
271
272
273
274
275
276
277
278
279
280
281
282
283
2. Stability
284
In order to anticipate stability issues, the Routh criterion
is applied to the denominator of the expression of the nor-
malized admittance, namely the coefficients (b3, b2, b1, b0)
of Eq. (8). By developing the Routh table,23one can obtain
the following parameters:
285
286
287
288
s3
s2
s1
s0
b3
b2
c1
c2
b1
b0
c3
c4
0
B
B
@
1
C
C
A;
(14)
289
where
c1¼ RmsðReþ RsÞ þLe
Cmcþ BlðBl þ CvÞ
?
MmsLe
Cmc MmsþRmsLe
c2¼ b0¼Reþ Rs
c3¼ c4¼ 0
ReþRs
hi
Cmc
8
>
>
>
>
>
>
>
>
>
>
:
>
<
>
>
>
>
>
>
>
>
>
:
(15)
290
The condition for the stability yields each coefficient from
the first column in the matrix of Eq. (14) should be of the
same sign, yielding
291
292
?Re< Rs
Cv>1
Bl
MmsLe
CmcMmsþRmsLe
½
ReþRs
??Le
Cmc? RmsðReþ RsÞ
??
? Bl
8
:
One can then observe that if the first condition is satisfied,
stability is always ensured if Cv? 0. This result is valid for
the ideal linear model of Sec. II A, but it should be well
understood that such a model does not account for various
phenomena that may have a prejudicial impact on stability,
such as the variation of electric resistivity and self-induct-
ance with frequency, or non-linear behavior (stiffness of the
suspensions induced by Laplace force), or even heat phe-
nomena occurring in the coil. Such phenomena are not easy
to model with accuracy but, in some cases, can be considered
in a lumped-elements model.24,25Nevertheless, the result of
Eq. (16) is considered in the following developments, with a
view of providing a criterion on stability for the ideal case.
<
:
(16)
293
294
295
296
297
298
299
300
301
302
303
304
305
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306 3. Equivalent electric load
307
308 ity V(s) and sound pressure Pþ(s) can be expressed as the
309 functions of the electric current I(s),
It also follows from Eq. (3) and Eq. (7) that both veloc-
EðsÞ ¼ CvVðsÞ þ CpPþðsÞ ? RsIðsÞ ¼ ?ZðsÞIðsÞ;
(17)
310 where Z(s) represents the equivalent electric load impedance,
311 ratio of the total control feedback voltage against current in-
312 tensity. This electric impedance becomes:
ZðsÞ ¼ ?ðsLeþ ReÞ
?
s2Leþ s
CpBl
Sþ Reþ Rs
SBlRms? 1 þCv
hi
s2Cp
SBlMmsþ s
Cp
Bl
??
hi
þ
Cp
SBlCmc
:(18)
313 Each of the control cases described hereafter is then equiva-
314 lent to an electrical network Z(s), which is composed of a
315 first negative series of resistance–inductance –Ze(s)¼
316 – (sLeþRe), which can be viewed as a “neutralization” of
317 the electric impedance of the loudspeaker, and a shunt im-
318 pedance Zs(s) that depends on the control case. This neutrali-
319 zation reveals the required electric network that, connected
320 to the loudspeaker, should fit the target acoustic admittance.
321 In this sense, this formulation can directly be used for syn-
322 thesizing electric networks capable of mimicking feedback-
323 based active absorption (e.g., Sec. III D). Conversely, each
324 shunt has its acoustic feedback counterpart, namely a setting
325 of the acoustic feedback gains Cpand Cvthat plays the same
326 role than the load impedance. Section III provides computa-
327 tional results, processed with the aforementioned formula-
328 tions of the acoustic admittance and equivalent electric
329 loads, for different examples of shunt and acoustic feedback
330 controls that can be covered by the denomination “electro–
331 acoustic absorber.”
332 III. CASE STUDY
333
In this section, the models are processed according to
334 the following assumptions:
335 (1)
336
An electrodynamic moving-coil loudspeaker (VisatonV
AL 170 low-mid-range loud speaker, the specifications
of which are given in Table I) is used as the EA.
R
337
338
339 340
(2)The rear face of the loudspeaker diaphragm is enclosed
in a box, the volume of which is Vb¼10l,
The loudspeaker front face is radiating at the termination
of a waveguide, the opposite extremity being considered
as perfectly absorbent.
341
342 343
(3)
344
345
346
The different settings considered in this section are
given in Table II. The acoustic absorption coefficients
obtained by simulations are gathered on the synthetic illus-
tration of Fig. 3, in order to show their common behavior
and assess the influence of the EA parameters on the acoustic
absorption coefficient on a single chart.
347
348
349
350
351
352
A. Case 0: Open-circuit
353
In the case where the electroacoustic transducer is not
connected to any electric load, no current is circulating in
the coil; thus no feedback force is created, and the device
can be described as “passive.” The acoustic admittance of
the passive diaphragm can then be written as:
s
s2MmsþsRmsþ
354
355
356
357
Y0ðsÞ ¼Zmc
1
Cmc;
(19)
358
This specific case provides insight on the behavior of other
shunt and feedback techniques, since the admittance of
359
TABLE I. Electroacoustic transducer small signal parameters considered
for the simulations.
ParameterNotationValueUnit
DC resistance
Voice coil inductance
Force factor
Moving mass
Mechanical resistance
Mechanical compliance
Effective area
Re
Le
Bl
Mms
Rms
Cms
S
5.6
0.9
6.9
15.0
0.92
1.2
133
X
mH
N A?1
g
N m?1s
Mm N?1
cm2
TABLE II. Examples of setting cases and corresponding control results.
Control settingsControl results
Rs
Cv
Cp
fea
(Hz)(X)(V m?1s)(V Pa?1)
fea
Qea
Case 0
Case 1
Case 2a
Case 2b
Case 2c
Case 3a
Case 3b
Case 3c
N=A
5
0
0
0
0
0
0
N=A
0
10
100
194
10
70.0
100
N=A
0
0
0
0
0.025
0.13
0.25
75.1
74.9
74.7
74.7
74.7
74.7
74.7
74.7
0.17
1.05
4.11
24.6
443.3
1.24
1.36
1.02
7.67
1.25
0.32
0.05
0.003
0.32
0.07
0.05
FIG. 3. Computed absorption coefficients of the EA for various setups (refer
Table II).
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Page 6

360 Eq. (11) is analog to Eq. (19). This expression exhibits a res-
361 onator behavior, the resonance frequency of which is deter-
362 mined by fEA;0¼fs¼1=ð2pÞ1=
363 being characterizedby
364 fEA;0¼ Rms=ðZmcÞ and the resonance quality factor being
365
QEA;0¼ 1=ðRmsÞ
366
The acoustic absorption coefficient of the open-circuit
367 EA is illustrated in Fig. 3, with the label “case 0,” presenting
368 a maximal value a0;max< 1 at the resonance. The “natural”
369 resonant behavior of the passive loudspeaker is highlighted,
370 with its relatively low capacity of absorbing the acoustic
371 energy around its resonance frequency, due to mismatched
372 mechanical losses (here Rms ? Zmc). This also indicates
373 that there is still some way to go before achieving total
374 absorption at resonance, which has to be done by electrically
375 adding losses in the system. This can be understood as an
376 underlying objective of shunt techniques, as described in
377 Sec. III B.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MmsCmc
the
p
, the total losses
normalizedresistance
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Mms=ðCmcÞ:
p
378 B. Case 1: Shunt resistance (including shortcut)
379
If we consider a single positive resistor Rsloading the
380 electric terminals of the loudspeaker, the normalized acous-
381 tic admittance can be expressed as follows:
Y1ðsÞ ? Zmc
s
s2Mmsþ s Rmsþ
ðBlÞ2
ReþRs
??
þ
1
Cmc
:
(20)
382 In this expression, the mechanical resistance of the trans-
383 ducer can be increased by ðBlÞ2=ðReþ RsÞ expressing addi-
384 tional losses in the electric circuit. As a consequence, the
385 absorption coefficient at the resonance can be easily varied,
386 so as to cater for different values from a0,maxup to 1. This
387 can also be explained by the fact that the positive electric re-
388 sistance, fed by the induced back electromotive force, cre-
389 ates an electrical current in the coil. This current generates a
390 feedback force at the loudspeaker diaphragm, modifying its
391 vibrating velocity in response to an exogenous sound pres-
392 sure. Thus, the loudspeaker connected to a passive shunt can
393 no longer be denoted as “passive,” as in the case of the
394 open-circuit, and rather calls for the label “semi-active,” due
395 to this intrinsic feedback force.
An optimal shunt value can be set so as to have a perfect
397 acoustic absorption at the transducer resonance,
396
Ropt¼
ðBlÞ2
Zmc? Rms? Re:
(21)
398 For the case of the studied electrodynamic loudspeaker in
399 the air at 20?C (q¼1.18 kgm?3and c¼340 ms?1), this
400 optimal resistance Roptis equal to 5 X. The corresponding
401 acoustic absorption coefficient is given in Fig. 3, with the
402 label “case 1.” It can be observed that the added resistance
403 allows a significative increase of the absorbing capability of
404 the loudspeaker over the bandwidth of interest to the point
405 where it is perfectly absorbent at resonance. This result is at
406 the heart of the principle of shunt loudspeakers,17the stabil-
407 ity of which is always ensured with passive dipoles. More-
408
over,
alternative to conventional sound absorbing materials in the
low-frequency range, which are often bulky and present poor
absorbing efficiency.2But a positive shunt only allows lim-
ited values of sound absorption, depending on the total resis-
tances of the loudspeaker system, and no substantial
broadening of the bandwidth of control is possible. The
intention of Secs. III C and III D is to further extend the pre-
ceding properties to active feedbacks, with an emphasis on
combined velocity=pressure-feedbacks.
thiselectroacousticsolutionisaninteresting
409
410
411
412
413
414
415
416
417
418
C. Case 2: Velocity-feedback—shunt negative
resistance
419
420
We consider the case where Rs¼0 X and Cp¼0
VPa?1, where the only diaphragm velocity feeds back the
electroacoustic transducer electrical input. The acoustic
absorption coefficient is computed after Eq. (6), the results
being reported on Fig. 3, with labels “case 2a” and “case
2b.” According to Eq. (13), the EA parameters become:
421
422
423
424
425
fEA;2¼
1
Zmc
RmsþBlðBl þ CvÞ
1
ReRmsþ BlðBl þ CvÞ
Re
??
ffiffiffiffiffiffiffiffiffi
QEA;2?
Mms
Cmc
r
8
>
>
>
>
:
<
:
(22)
426
The values of the equivalent mechanical resistance can then
be set from a constant value Rmsþ ðBlÞ2=Re, which depends
on the loudspeaker’s passive resistances and corresponds to
the short-circuit case, up to infinity in theory. In the case of
high velocity-feedback gains, the control forces the dia-
phragm to be ideally rigid, corresponding to perfect reflec-
tion. Moreover, as feedback gain increases, the quality factor
decreases in proportion leading to a consecutive broadening
of the control bandwidth.
It is also noticeable that according to Eq. (18), the
equivalent shunt of the velocity-feedback is a negative im-
pedance, when Cvtakes positive values. Such velocity-feed-
back, whatever the means to sense velocity, then consists in
applying a negative impedance circuit at the transducer elec-
trical terminals, such that:
427
428
429
430
431
432
433
434
435
436
437
438
439
440
Z2ðsÞ ¼ ?
Cv
Cvþ BlðsLeþ ReÞ:
(23)
441
This negative impedance can be obtained with a Wheatston
bridge loading the electrical terminals of the loudspeaker, as
reported in Ref. 13, confirming the possibility to design EAs
capable of self-sensing acoustic quantities out of dedicated
electric filters. The stability of velocity-feedback control is
theoretically ensured if Cvis positive, according to Eq. (16).
In practice though, the setting of the negative impedance is
very sensitive to the values of resistance and inductance. An
actual limitation of gains is encountered, mainly due to the
variation of Leand Rewith frequency that are not considered
in this model. However, as shown in Sec. IV, there is still a
large margin of gains to achieve velocity-feedback without
facing instability of the device.
442
443
444
445
446
447
448
449
450
451
452
453
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454 D. Case 3: Direct impedance control
455
“Direct impedance control”6refers to the combination of
456 a velocity- and a pressure-feedback at the loudspeaker electri-
457 cal terminals. In this case, the normalized acoustic admittance
458 Y3takes the general form of Eq. (8) with Rs¼0 X, and
fEA;3?Bl þ Cv
ZcCp
Zc
1
ReRmsþ BlðBl þ CvÞ
?1
Cv
Cp
QEA;3?
ffiffiffiffiffiffiffiffiffi
Mms
Cmc
r
:
8
>
>
>
>
:
<
(24)
459 This
460 Cv ? Bl, the target acoustic resistance is directly accessi-
461 ble through the ratio Cv=Cpthat should equal the character-
462 istic medium impedance Zc¼401.2 kgm?1s?1(in the air
463 at 20?C) in view of the total absorption. As for velocity-
464 feedback, the extension of the control bandwidth is made
465 possible by increasing gain Cv, theoretically up to infinity,
466 which is hardly the case in practice for the same stability
467 reasons as in the preceding example. The setting of the EA
468 appears quite straightforward, consisting of first adjusting
469 the ratio of gains to equal a desired acoustic resistance
470 value, and then increasing simultaneously the two gains
471 (while their ratio remains constant) up to the above-men-
472 tioned instability threshold.
The results given in Fig. 3 with label “case 3a” and
474 “case 3c” demonstrate the possibility of achieving wideband
475 acoustic absorption with such an EA: In these cases, the
476 obtained acoustic impedance matches the target resistance
477 Zc¼ Cv
478 value of Cv(decreasing quality factor).
479
Moreover, the equivalent electric shunt of Eq. (18) is
480 processed with the parameters of case 3a, leading to the
481 function of Eq. (25) illustrated in Fig. 5(a).
resultindicatesthatassumingfeedbackgain
473
?Cp on a frequency bandwidth increasing with the
Z3aðsÞ þ ðsLeþ ReÞ ¼
0:00041s2þ 8:5s
?0:0019s2þ s ? 416:44:
(25)
482 This target electric impedance can be obtained with the
483 electric network illustrated in Fig. 4, composed of electric
484 resistances (R1 and R2) and inductances (L1 and L2).
485 Here, Zsdenotes the left part of the electric shunt, exclud-
486 ing the neutralizing electric impedance –(sLeþRe). In
487 this case,
ZsðsÞ ¼
s2L2 1 þR1
R2þ s 1 þL2
R2
?
??
1 þR1
þ sR1
s2L2
L1
R2
?hi
þR1
L1
:
(26)
488
The identification of the parameters of this network is not
straightforward and requires much care, since the number of
degrees-of-freedom (R1and R2, L1and L2) is lower than the
number of coefficients of the target electric impedance.
489
490
491
FIG. 4. Example of electric impedance synthe-
sis as an active shunt of an EA.
FIG. 5. (a) Simulation of the target electric impedance of Eq. (25) (square
markers), and the synthesized electric network impedance of Eq. (28) (round
markers) for the “case 3a” (plain lines, real part; dotted lines, imaginary part);
(b) Simulation of the acoustic absorption coefficient corresponding to case 3a,
with the two equivalent methods (plain line, with direct impedance control;
square markers, with the synthesized shunt electric network of Fig. 4).
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Page 8

492 Nevertheless, illustrating the equivalence between shunt and
493 feedback control, a set of electric components has been cho-
494 sen, so that the coefficient of s2on the numerator of Eq. (26)
495 equals 0, or in other words, the synthesized impedance fits the
496 target one in the low-medium frequency range. It yields,
R1¼ ?R2¼ 8:5X; L1¼ ?18:7mH; L2¼ 17:4mH(27)
497 By replacing the values of (R1, R2, L1, L2) in Eq. (26), one
498 can obtain the following synthesized impedance:
ZsðsÞ ¼
8:5s
?0:0020s2þ s ? 454:5;
(28)
499 that can be compared to the expression of Eq. (25), as illus-
500 trated in Fig. 5(a). Thus, the synthesized electric impedance
501 matches the target one within the frequency bandwidth of
502 interest.
Conversely, this synthesized electric impedance Zs(s)
504 forms a new shunt impedance in series with –(sLeþRe) that
505 we can then substitute for Rsin Eq. (7) [or even Eq. (20)] to
506 compute the corresponding “synthesized” acoustic admit-
507 tance denoted by Ys,3a(in order to distinguish this synthe-
508 sized admittance and the target one Y3a). The synthesized
509 acoustic admittance is then:
503
Ys;3aðsÞ¼
Zmc
s
s2MmsþðBlÞ2 L2
R1R2
??
þs RmsþðBlÞ2
R1
??
þ
1
CmcþðBlÞ2
L1
??
(29)
510 This normalized admittance, with the chosen values of
511 Eq. (27), corresponds to a theoretically stable configuration
512 of the EA, according to the Routh criterion, assuming that
513 the neutralization of the electric impedance is ideally
514 achieved. The “synthesized” absorption coefficient is then
515 processed, according to Eqs. (6) and (29), and compared in
516 Fig. 5(b) to the one obtained with direct impedance control
517 (with Cv¼10 Vm?1s, Cp¼0.025 VPa?1, Rs¼0 X).
518
This last result illustrates the formal equivalence
519 between shunt loudspeakers and feedback-based active
520 sound absorption showing similar results in terms of sound
521 absorption. One can observe that, with the chosen electric
522 network, the coefficients of s2and s0in the denominator of
523 the synthesized acoustic admittance Ys,3a(s) are lower than in
524 the passive case [see Eq. (19)]. This is in accordance with
525 the objective of lowering the equivalent mass and increasing
526 the equivalent compliance of the loudspeaker in order to
527 extend the bandwidth of the control. Moreover, the equiva-
528 lent acoustic resistance is actually higher than the passive
529 one, which is required to match the acoustic resistance of air.
530 The electrical network allows the adjustment of the three pa-
531 rameters of the acoustic resonator to the target. This result
532 opens the way to new strategies for the optimization of elec-
533 tric networks shunting a loudspeaker in view of active sound
534 absorption. Practically, such electric impedance design is not
535 straightforward and needs a very accurate selection of the
536 electric components, especially with respect to stability, but
537 also in terms of absorption performances. The implementa-
538
tion of such impedance synthesis strategy on digital signal
processing platforms could help alleviate these issues, but
these developments are out of the scope of this paper.
539
540
541
E. Discussions
542
The results in Fig. 3 (see Table II for control parame-
ters) clearly highlight the similarities between the different
control techniques detailed in Secs. III A-D, unifying passive
shunt techniques and active feedback control of acoustic im-
pedance into a single formalism. The passive performances
of an EA can be first improved with a simple passive electric
resistance of optimal value so as to reach almost perfect
absorption within a narrow frequency bandwidth around the
resonance due to the increase of total resistances and the
slight decrease of the resonance quality factor. The band-
width of control can then be significantly increased by
choosing appropriate feedback gains in a combined pres-
sure–velocity-feedback. This leads to an enhanced damping
at resonance, and a highly decreased quality factor of reso-
nance, resulting from the lowering of the apparent mass and
compliance of the resonator. Inspired by these formal analo-
gies, a technique for adjusting the active feedback gains is
also introduced. It consists in tuning the three independent
parameters of the equivalent acoustic resonator, presenting
interesting perspectives for controlling the acoustic imped-
ance of an electroacoustic loudspeaker. As long as the con-
sidered lumped-element model is valid [at least up to the
cut-off frequencies of Eq. (10)], a criterion for stability has
been identified allowing a wide range of settings for the reso-
nator parameters. On the other hand, the acoustic performan-
cesobtainedwithsynthesized
theoretically match the ones obtained with pure acoustic
feedbacks. The synthesis of electric impedances is also
shown to be theoretically possible with quite simple electric
networks, such as the one of Fig. 4, assuming that the neu-
tralization of the loudspeaker electric impedance has been
primarily performed with accuracy. But, in practice, this is
very sensitive to the electric components, and stable analogi-
cal implementations have been difficult to be realized at this
stage. However, this theoretical result still presents interest-
ing properties of EAs and bridges a conceptual gap between
shunt loudspeakers and feedback-based active acoustic im-
pedance control. Section IV intends to experimentally vali-
date the aforementioned properties.
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
electricimpedances
568
569
570
571
572
573
574
575
576
577
578
579
580
581
IV. EXPERIMENTAL ASSESSMENT
582
In order to assess experimentally the equivalence
between the active feedback control and electric shunts, a
closed-box (volume Vb¼10 l) VisatonV
range loudspeaker is employed as an EA. The acoustic
absorption coefficient of the EA is assessed after ISO 10534-2
standard,26as described in Fig. 6. In this setup, an imped-
ance tube is specifically designed (length L¼3.4 m; internal
diameter ؼ150 mm), one termination of which is closed
by an EA, the other extremity being open with a horn-shape
termination so as to exhibit anechoic conditions.27A source
loudspeaker is wall-mounted close to this termination. Two
holes located at positions x1¼0.46 m and x2¼0.35 m from
583
584
RAL 170 low-mid-
585
586
587
588
589
590
591
592
593
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Page 9

594 the EA position are the receptacles of 1=200microphones
595 (Norsonic Type 1225 cartridges mounted on Norsonic Type
596 1201 amplifiers), sensing sound pressures p1¼p(x1) and
597 p2¼p(x2) and the transfer function H12¼ p2=ðp1Þ is proc-
598 essed through a 01dB-NetdB Multichannel Analyzer. Simul-
599 taneously, with a view to process the equivalent electric load
600 Z at the EA electric terminals, the electric voltage e and cur-
601 rent i circulating through the coil are measured and proc-
602 essed with the same instrumentation.
In this experimental study, the active feedback settings
604 corresponding to case 2c and case 3b have been applied at
605 the EA electric terminals, as well as at the optimal shunt of
606 case 1. Here, the velocity-feedback is processed through a
607 Polytec OFV-505=5000 laser velocimeter (sensitivity being
608 set to rv¼100 Vm?1s). This velocity sensor is positioned
609 at the output of the open tube, as illustrated in Fig. 6—the
610 laser beam focusing on a single point of the radiator at the
611 middle of its radius. The pressure is sensed with an external
612 PCB 130D20 microphone (sensitivity of rp¼47.5 mVPa?1)
613 located in the plane x¼0 and slightly off-center (at a height
614 of z¼3.2 cm from the duct wall), yielding a distance of
615 approximately 5 mm from the loudspeaker diaphragm. The
616 direct impedance control is processed through a two-way an-
617 alogical audio-mixer, allowing the setting of electric feed-
618 back gains C0
619 absorption coefficients measured with the above-mentioned
620 setup are compared to the corresponding model simulations
621 described in Sec. III and illustrated in Fig. 7(a). In parallel,
622 the equivalent electric load, processed as the transfer func-
623 tion between voltage e and current i at the EA terminals, is
624 assessed for the settings of case 3b and compared to the cor-
625 responding model simulation given in Fig. 7(b).
The absorption coefficient presented at the front face of
627 the loudspeaker can be easily varied from almost total reflec-
628 tion up to total absorption over a wide frequency bandwidth
629 depending on the control case. The theoretical curves show
630 the same trend as the experimental values even if some slight
631 differences can be observed with the ISO 10534-2 technique.
632 These measurements also show a similar behavior of differ-
633 ent acoustic resonators, presenting variable acoustic resistan-
603
v¼ Cv=ðrvÞ and C0
p¼ Cp
?ðrpÞ. The sound
626
634
ces and quality factors, given in Table II. With such
formalism, wide-band acoustic absorbers can be designed in
a very straightforward manner according to certain specifica-
tions: If an application requires narrow-band absorption,
635
636
637
FIG. 6. Experimental setup for
the assessment of EAs absorption
Coefficient.
FIG. 7. Experimental assessment of the EA and comparison to numerical
simulations: (a) absorption coefficient obtained with optimal shunt (case 1:
Rs¼5 X) velocity-feedback (case 2c: Cv¼194 Vm?1s) and direct imped-
ance control (case 3b: Cv¼70.0 Vm?1s and Cp¼0.13 VPa?1); (b) meas-
ured and simulated equivalent electric load for case 3b.
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Page 10

638 passive shunt can easily be deployed, and in other cases sta-
639 ble and still simple active impedance control can readily be
640 set. Moreover, the electric assessment confirms that, when
641 connected to an active feedback control device, the EA
642 behaves as if it was connected to an active electric load, the
643 elements of which can be identified on the measured transfer
644 function. These assessed values also match the transfer func-
645 tion of the electric network given as example in Sec. III D.
646 Slight discrepancies are observed between the theory and ex-
647 perimental data, confirming the importance of accurately
648 identifying the components of the loudspeaker to be
649 employed in the active device. For example, it appears in
650 Fig. 7(b) that the dependence of Reand Lewith frequency
651 should be accurately modeled with a view to synthesizing
652 the requested electric network.
On the stability side, all the active control gains pre-
654 sented in this paper have been set under the threshold of
655 instability. In practice, instability can be experienced while
656 further increasing the feedback gains, the threshold depend-
657 ing on the reactive components in the electroacoustic loud-
658 speaker (especially the electric inductance), as explained in
659 Sec. II B 2. As an illustration, the loop stability is assessed
660 for the settings of case 3b, according to the stability criterion
661 presented in Ref. 28 for single channel feedback control.
662 This argument says that, if the phase of the transfer function
663 between the input and the output at the disconnection in the
664 loop is 360?, then the magnitude of the transfer function
665 should be less than unity. Here, the open-loop gain is meas-
666 ured with the 01dB-NetdB Multichannel Analyzer process-
667 ing the transfer function between the signals provided at the
668 input of the power amplifier feeding the EA and the output
669 of the audio-mixer. This open-loop gain is first measured
670 when the EA is placed at the entrance of the impedance tube
671 in Fig. 7, and in a second step in the case where it is moved
672 to an anechoic chamber (free-field conditions). The compari-
673 son is illustrated in Fig. 8. The results are presented on an
674 extended bandwidth (10–5000 Hz) to identify potential high-
675 frequency effects.
Generally speaking, the main problem might arise at the
677 resonance frequency of the EA, where the open-loop gain
678 shows the highest magnitude. This problem is easily allevi-
679 ated by choosing appropriate settings for the two feedback
680 gains. Indeed, for a certain target acoustic impedance
681 Cv=ðCpÞ, there always exists a combination of the two feed-
653
676
682
back gains Cvand Cpyielding a 180?phase rotation at this
frequency. Apart from this problem, stability issues could
also arise from the reactive component of the EA, such as
the electric inductance of the coil, or higher-order resonan-
ces of the diaphragm, as can be observed above 1 kHz on the
free-field measurement. These problems might generally
represent limitations for setting the feedback gains, but do
not actually affect the acoustic performances of the device in
the frequency bandwidth of interest.
However, the main instability issues occurring during
the assessments appear to be related to the experimental fa-
cility. In the impedance tube especially, the open-loop gain
illustrated in Fig. 8 presents a gain margin of about 2 dB (at
2641 Hz) and a phase margin of 2.7?(at 4781 Hz). One can
observe that these margins are significantly increased in the
case where the acoustic absorber is in a free-field environ-
ment; the resonances of the impedance tube represent the
most important factor of magnitude and phase variations in
the open-loop gain. As a conclusion, the observed instability
issues should not be entirely taken for an intrinsic property
of the EA, and the margins for setting the EAs are actually
much higher than the ones assessed in the impedance tube.
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
V. CONCLUSIONS
705
A unifying theory of active acoustic impedance control
has been introduced, covering different control techniques
from passive shunt to pressure=velocity-feedbacks in a sin-
gle formalism. An acoustic feedback is shown to be equiva-
lent to an electrical load at the transducer electrical
terminals. Conversely, a synthetic electric network has been
identified for each active acoustic impedance control, the
design of which can be specified in a relatively simple man-
ner. Broadband acoustic performances have been measured
on a generic prototype of EA with passive shunt and active
acoustic feedback control. The tested configurations present
a variety of acoustic absorption, which are in good agree-
ment with the simulations. Finally, the equivalent electric
load of an active acoustic feedback has also been experimen-
tally assessed, confirming the theory of electroacoustic
absorbers.
Further work is ongoing, focusing on the design and
optimization of dedicated electrical networks. The optimiza-
tion addresses the identification of the target electric
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
FIG. 8. Experimental assessment of
the open-loop gain (left upper chart,
magnitude in decibel; left lower
chart, phase in radians; right chart,
Nyquist plot) for the direct imped-
ance control setting (case 3b) in dif-
ferent acoustic environments (plain
lines, in the impedance tube; dotted
lines, in the anechoic chamber).
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724 impedance expressed in Eq. (26), along the same methodol-
725 ogy as the one reported for optimizing semi-active shunt
726 loudspeakers.29A potential application of the reported con-
727 cepts could consist in designing specified electric filters ca-
728 pable of sensing acoustic quantities (pressure and=or
729 diaphragm velocity) out of electrical current=voltage, thus
730 preventing the use of external sensors in active noise control
731 devices.
732 ACKNOWLEDGMENTS
733
This work was supported by the Swiss National Science
734 Foundation under Research Grant No. 200021-116977. The
735 authors also wish to thank Patrick Roe and Travis Forbes for
736 their helpful reviewing of the manuscript.
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
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753
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AQ2
control system for noise reduc-
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810 AQ1: Please check references 9 and 20. The recorded titles for these patents do not match the actual titles in the WIPO data-
811 base: Ref. 9 lines 757 to 760: WO 1997/003536) LOUDSPEAKER CIRCUIT WITH MEANS FOR MONITORING THE
812 PRESSURE AT THE SPEAKER DIAPHRAGM, MEANS FOR MONITORING THE VELOCITY OF THE SPEAKER DIA-
813 PHRAGM AND A FEEDBACK CIRCUIT
814 AQ2: Ref. 20 lines 786 to 787: (WO 1999/059377) Active Acoustic Impedance Control Device