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LIBRARY

Michigan State
Unlvers “V

 

 

 

 

This is to certify that the

dissertation entitled

DEVELOPMENT OF AN ACTIVE ACOUSTIC SINK (MS)

FOR NOISE CONTROL IN ACOUSTIC SPACES
presented by

SACHIN D . GOGATE

has been accepted towards fulfillment
of the requirements for

Ph. D. Mechanical Engineering

degree in

 

 

 

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINE return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE I DATE DUE DATE DUE

5M mesh

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1/93 chan.p65—p.14

DEVELOPMENT OF AN ACTIVE ACOUSTIC SINK (AAS)
FOR NOISE CONTROL IN ACOUSTIC SPACES

By
Sachin D. Gogate

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1995

 

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330i}

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ABSTRACT

DEVELOPMENT OF AN ACTIVE ACOUSTIC SINK (AAS)
FOR NOISE CONTROL IN ACOUSTIC SPACES

By
Sachin D. Gogate

Active control of undesirable acoustic noise satisifies one of the two distinct
acoustic design objectives : sound cancellation and sound power absorption. Sound
cancellation has been traditionally used for control of acoustic noise. It works on the
principle of destructive interference of sound fields and is achieved by superimposing an
original sound field with an electroacoustically generated sound field of opposite phase.
Sound power absorption is achieved by removing acoustic energy due to noise from an
enclosed space.

Active Acoustic Sink (AAS), an active noise control system based on sound power
absorption is developed in this work. It uses speaker to absorb acoustic energy from an
acoustic volume. The speaker velocity must track speaker face acoustic pressure to ensure
acoustic energy flow in the speaker.

The three stages of research discussed in this work are speaker dynamic modeling,
speaker compensation design, and AAS controller design. In the first stage, a
methodology for constructing a speaker model is developed and experimentally verified
on a dual-wound coil subwoofer. Velocity feedback compensation for the subwoofer is
then designed in the next stage to ensure speaker cone velocity tracking of speaker face
pressure. The AAS controller is designed, implemented, and tested in the final stage.

Experimental tests of AAS indicate successful performance.

 

To Sou. Aai.

and advice, 1

Dedication

To Sou. Aai, Ti. Dada, Sou. Shubhatai and Suchatai without whose help, encouragement,

and advice, I would not have come this far.

iv

 

Copyright

Sachin D.

1995

Copyright

Copyright by

Sachin D. Gogate

1995

 

”J

 

I wm
not only his
also like to I.
have been im

I WOL
FitzSimons, .
during my stu

I woul

during my Stu.
lgratel

the fl.‘(tures ne.
1 Woult

Smiay Mama.
{ht darkest rim,
been [echm'CaL
friends, Dr. St:
Pmad- Nandu.
I Would

Ruth Andersland

Acknowledgments

I would like to express my sincere appreciation to Professor Clark J. Radcliffe for
not only his technical advice but also for his valuable tips on professional skills. I would
also like to thank him for the generous financial support without which this study would
have been impossible.

I would like to thank my committee members, Professors C. MacCluer, P.M.
FitzSimons, A. G. Haddow and R. C. Rosenberg for their advice, comments and help
during my studies here.

I would like to thank my parents and sisters for their encouragement and support
during my studies at Michigan State University.

I gratefully acknowledge the assistance of Bob Rose and Leonard Eisele who made
the fixtures necessary for the experiments in this work.

I would like to give special thanks to my roommates at MSU, Manoj Thatte and
Sanjay Mishra. I cannot express in words their valuable contribution and support during
the darkest times of dissertation. Although the support they have given me has not always
been technical, I feel that it has made me a better person. Thanks also go to my dear
friends, Dr. Sanjeev Kholkute, Shanti Vedula, Uma Iyer, Hari, Yogesh Pathak, Shail,
Prasad, Nandu, Venky and Pratima. You all have made my life at MSU an enjoyable one.

I would like to thank my officemate, XianLi Huang for his friendship and help
over the years. I would also like to thank my friends, Jerome Palazzolo, Byam Brooks,

Ruth Andersland, and Chang-p0 Chao for making my stay at MSU enjoyable.

LIST OF TA]
LIST OF FIG
Chapter 1 In
1.1 O'
1.2 Pr
1.3 A
1.4 C!
Chapter ‘2 1,-
2.1 Ir

2.2 .‘v

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES ......................................................................................................... ix
Chapter 1 Introduction ................................................................................................... 1
1.1 Overview ........................................................................................................ 1
1.2 Previous Work ............................................................................................... 1
1.3 A Novel Noise Control Technique ................................................................ 5
1.4 Chapter Summary .......................................................................................... 7
Chapter 2 Identification and Modeling of Speaker Dynamics ...................................... 9
2.1 Introduction .................................................................................................... 9
2.2 Modeling a Dual-Wound Coil Subwoofer Speaker ..................... . .................. 9
2.3 Speaker Transfer Functions ........................................................................... 14
2.4 Speaker Parameter Identification Procedure ................................................. 15
2.5 Parameter Identification of a Dual-Wound Coil Subwoofer Speaker ........... 17
2.6 Speaker Enclosure Effects ............................................................................. 20
2.7 Summary ........................................................................................................ 23
Chapter 3 Velocity Feedback Compensation of Speakers ............................................. 25
3.1 Introduction .................................................................................................... 25
3.2 Speaker Velocity Feedback Design ............................................................... 26
3.3 Speaker Cone Velocity Sensor Design .......................................................... 27
3.4 Stability Analysis of Speaker Velocity Feedback System ............................. 32
3.5 Performance Evaluation of Speaker Velocity Feedback ............................... 34
3.6 Summary ........................................................................................................ 37

vi

 

Chapter-I [5
4.1 Ir
4.2 T.
4.3 A
4.4 Sr
4.5 St
ChapterS 11‘.
5.1 In
5.2 In‘
5.3 Pc
5.4 Su
Chapter6 Q
6.1 Di:

6.2 Di]
BIBLIOGRA]

Chapter 4 Active Acoustic Sink Control Model Development ..................................... 39

4.1 Introduction .................................................................................................... 39
4.2 Theory of Active Acoustic Sink System ....................................................... 39
4.3 Active Acoustic Sink Controller Design .................................... 43
4.4 Stability Analysis of Active Acoustic Sink System ...................................... 45
4.5 Summary ........................................................................................................ 48
Chapter 5 Implementation and Testing of Active Acoustic Sink Controller ................. 49
5.1 Introduction .................................................................................................... 49
5.2 Implemetation of Active Acoustic Sink Controller ....................................... 49
5.3 Performance Evaluation of Active Acoustic Sink System ............................ 51
5.4 Summary ........................................................................................................ 60
Chapter 6 Conclusions ................................................................................................... 62
6.1 Dissertation Summary ................................................................................... 62
6.2 Directions for Future Work ........................................................................... 64
BIBLIOGRAPHY ............................................................................................................ 65

vii

 

 

Table 1.1 lde
Table 5.1 Ac

Table 5.2 lncr

811111

LIST OF TABLES

Table 1.1 Identified Parameters for a Dual-Wound Subwoofer Speaker ........................ 18

Table 5.1 Acoustic Energy absorbed by the Active Acoustic Sink ................................ 59

Table 5.2 Increase in Absorbed Acoustic Intensity from Uncontrolled to Controlled

States ............................................................................................................... 6O

viii

 

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without

 

LIST OF FIGURES

Figure 1.1 Minimum Effective Frequency for Polyurethane Foam and Fibrous

Glass at Different Thickness .......................................................................... 2
Figure 1.2 Diagrams Describing Paul Leug's Patent on Active Noise Cancellation ....... 3
Figure 1.3 The Active Sound Cancellation Device constructed by Olson and May ....... 4

Figure 1.4 Manually Adaptive Active Noise Cancellation System proposed by

Conover .......................................................................................................... 4
Figure 1.5 Schematic of the Active Acoustic Sink System ............................................. 6

Figure 2.1 Radio Shack Realistic 12 inch, Dual-Wound, Subwoofer Model 40-
1350 ............................................................................................................... 10

Figure 2.2 Schematic Representation of a Dual-Wound Coil Subwoofer Speaker ......... 10

Figure 2.3 Bond Graph Model of a Dual-Wound Coil Subwoofer Speaker without

Enclosure ....................................................................................................... 1 1

Figure 2.4 Schematic of Experimental Setup for Measuring Speaker Velocity

Response ........................................................................................................ 17

Figure 2.5 Comparison of Measured and Speaker Model Velocity Responses
without Speaker Coil Inductance (dB re 1) ................................................... 19

Figure 2.6 Comparison of Measured and Speaker Model Velocity Responses with

1X

 

 

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Er.

Figure 2.8 Co

Re

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Speaker Coil Inductance (dB re 1) ................................................................. 20

Figure 2.7 Bond Graph Model of Dual-Wound Coil Subwoofer Speaker with

Enclosure ....................................................................................................... 21

Figure 2.8 Comparison of Measured and Enclosed Speaker Model Velocity
Response (dB re 1) ......................................................................................... 23

Figure 3.1 Schematic of Speaker Velocity Feedback Compensation .............................. 27

Figure 3.2 Secondary Coil Voltage/Speaker Cone Velocity Model vs. Measured
Responses (dB re 1) ....................................................................................... 30

Figure 3.3 Compensated Secondary Speaker Coil Voltage/Laser Velocity
Response (dB re 1) ......................................................................................... 31

Figure 3.4 Measured Bode Plot of the Open-Loop Transfer Function of Speaker
Velocity Feedback System (dB re 1) ............................................................. 33

Figure 3.5 Root Locus of Velocity Feedback System for Varying Proportional

Controller Gain .............................................................................................. 34
Figure 3.6 Experimental Setup for Speaker Velocity Feedback ...................................... 35

Figure 3.7 Comparison of Measured and Model Closed-Loop Speaker Velocity
Response for Proportional Gain of K p =10 (dB re 1) ................................. 36

Figure 3.8 Comparison of Measured and Model Closed-Loop Speaker Velocity
Response for Proportional Gain of K p = 50 (dB re 1) ................................. 36

Figure 3.9 Comparison of Measured and Model Closed-Loop Speaker Velocity
Response for Proportional Gain of K p = 100 (dB re 1) ............................... 36

 

 

Figure 5.1 Open-

rrgire 5.2 Bode 1

 

0.004»
Figure 5.3 Phase I
Speake

Figure 5.4 Active

Figure 5.5 Speak:

Figure 5.6 Speuke

Figure 5.7 Measu.

PM 5-8 Acousr

5121168 .

Figure 3.10 Comparison of Measured and Model Closed-Loop Speaker Velocity

Response for Proportional Gain of K p = 200 (dB re 1) ............................... 37
Figure 4.1 Control Volume containing Idealized Point Flow Source ............................. 40
Figure 4.2 Block Diagram of Active Acoustic Sink ........................................................ 44

Figure 4.3 Comparison of Theoretical and Measured Magnitude Response of

Room Acoustics (dB re 1 Pa-scc/m) .............................................................. 48
Figure 5.1 Open-Loop Configuration of Active Acoustic Sink System .......................... 50

Figure 5.2 Bode Diagram of Open Loop Active Acoustic Sink System (dB re
0.004) ............................................................................................................. 50

Figure 5.3 Phase Shift, A¢x , due to Finite Distance between Microphone and

Speaker Cone ................................................................................................. 52
Figure 5 .4 Active Acoustic Sink Experimental Setup ..................................................... 54
Figure 5.5 Speaker Impedance of Uncontrolled AAS System (dB re 1 Pa-sec/m3) ....... 54
Figure 5.6 Speaker Impedance of Controlled AAS System (dB re 1 Pa—sec/m3) ........... 55
Figure 5.7 Measured Coherence in the Open Loop Frequency Response of AAS ......... 57

Figure 5.8 Acoustic Intensity Absorbed by AAS in uncontrolled and controlled

states .............................................................................................................. 59

xi

 

 

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Chapter 1 Introduction

1.1. Overview

Noise control can produce quiet environments that are comfortable and productive.
High noise levels in work areas are perceived as indicators of an inferior working
environment. The reduction of undesirable noise has been gaining attention over recent
years as people are becoming more and more noise conscious. The reduction of noise is

often necessary in power plants, factories and passenger compartment of vehicles.

1.2. Previous work

Control of acoustic noise is conventionally achieved by either passive or active
methods. Passive noise control using sound absorptive material (Crocker, et al., 1982,
Shuku, et a1. 1972, Ling, 1993) does not work well at low frequencies where the sound
wavelength is large compared to the treatment thickness. Figure 1.1 shows the minimum
effective frequency for absorptive materials at different thicknesses. The computation is
done based on half-power point frequencies of material acoustic absorption coefficients
(Crocker et al., 1982). The sound control effectiveness of a 2 inch thick absorption
material is quite limited below 400 Hz and decreases rapidly with decreasing frequency.
Small structural velocities at low frequencies also limit the effectiveness of passive

structural damping treatments (Joachim et al., 1981).

 

Frequency, f (”7.)

Figure 1.1.3

Different Ti

35

 

 

 

 

 

 

 

 

 

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8 15
cr
8
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\
5 fiberin SS
-— —_ I - ’

 

 

 

 

 

 

 

 

 

 

0 0.5 1 1.5 2 2.5 3 3 .5 4
Absorption Material Thickness, t (in)

Figure 1.1. Minimum Effective Frequency for Polyurethane Foam and Fibrous Glass at
Different Thickness (From "Noise and Vibration Control" by M.J. Crocker, and RM.
Kessler, Vol. II, 1982, pg. 23.

Active noise control using noise cancellation is achieved by superimposing the
original primary sound field with an electroacoustically generated secondary field of
opposite phase. It works on the principle of destructive sound fields. German inventor
Paul Leug was the first person to describe the idea of sound cancellation in his patent
published in United States in 1936 (Figure 1.2). A historical discussion of Leug's work
has been presented by Guicking (1990). According to his principle, the unwanted sound is
picked up by one or more microphones, their electrical signal fed, after amplification, to
one or more loudspeakers so that the sound wave produced is in opposing phase to the
primary unwanted sound and cancels it. Olson and May (1953) described a similar active
noise cancellation system which they misleadingly called an electronic sound absorber.

They described a system for actively canceling the sound detected by a microphone, which

2

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is placed clos:
time as 0150
acoustic noise
at even harmc
aharrnonic sc
the amplitude

particular diret

Figure 1.2. Di

AC0;

is placed close to a loudspeaker acting as a secondary source (Figure 1.3). About the same
time as Olson and May, William Conover (1956) demonstrated active reduction of
acoustic noise from large main transformers. The sound radiated by these transformers is
at even harmonics of the line frequency. Conover in his control system (Figure 1.4) used
a harmonic source generated from the transformer's line voltage. He manually adjusted
the amplitude and phase of the harmonic source signal to cancel the acoustic pressure in a

particular direction away from the transformer.

"I“ Intel 0. 1934
fig I

 

 

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Figure 1.2. Diagrams Describing Paul Leug's Patent on Active Noise Cancellation (From
"On the invention of active noise control by Paul Leug", by D. Guicking, Journal of

Acoustical Society of America, Vol. 87, No. 5, 1990, pp. 2251-2253)

 

i
“‘7

 

“.4“...

Figure 1.3. T

"Electronic S

 

 

AIPLIFIEI

 
    

 

 

 

 

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LOUDSPEAKER

CABINET AISOIIINC
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Figure 1.3. The Active Sound Cancellation Device constructed by Olson and May (From
"Electronic Sound Absorber", by HP. Olson and E.G. May, Journal of Acoustical Society
of America, Vol. 25., No.6, 1953, pp. 1130-1136)

W... w , Di

‘ souuo ‘ souuo me .
}. Lunch
11.55%: manna \ \ ,

 

 

 

 

 

 

 

 

 

 

Figure 1.4. Manually Adaptive Active Noise Cancellation System proposed by Conover
(From "Fighting Noise with Noise", by W.B. Conover, Noise Control 2, 1956, pp. 78-82)

The advent of digital signal processing techniques and devices has made it possible
to realize these noise cancellation algorithms in practice. A number of active noise control
systems based on the principle of sound cancellation have been developed for vehicle

interior noise control (Costin etal., 1989, Elliot et al., 1990, Sutton et al., 1990, Bao eta1.,

1991) and duct noise control (Manjal and Eriksson, 1988, Eriksson et al., 1988,
Swinbanks, 1973) These control systems use sensor microphones for measurement of
acoustic noise and generate "antinoise" through audio speakers driven by an adaptive
algorithm. The "antinoise" cancels the correlated acoustic noise at measurement
microphone locations. The acoustical effect of noise cancellation at microphones is to
generate "zones of silence" around these locations. It has been observed that the spatial
extent of "zones of silence" is of the order of one-tenth of an acoustic wavelength (3.4 m
at 100 Hz) and rapidly decreases as the frequency of noise increases above 200 Hz (Elliot
et al., 1988). Beyond these "zones of silence", sound cancellation may even increase the

resulting noise at some locations.

1.3. A Novel Noise Control Technique

Active Acoustic Sink (AAS), an active noise control system, based on the
objective of sound power absorption is proposed in this work (Radcliffe et al. 1994, Hall,
1994)). Sound power is the rate at which energy is radiated and is described by sound
intensity which measures energy flow per unit area. The AAS (Figure 1.1) will use an

electromechanical speaker as a control actuator to absorb acoustic energy. The

instantaneous acoustic intensity, Ii , at the speaker face is the product of acoustic pressure,
p, and particle velocity, v.

1
Ii = pv = _—Aspkr pfaceqspkr (1.1)

Here the acoustic pressure is equated to speaker face pressure, p face, and acoustic particle
velocity is related to speaker volumetric flow rate, qspkr, and the effective area of the
speaker, Aspkr- A microphone placed at the speaker face measures the total acoustic

pressure, Pface- The AAS theory for absorption of acoustic energy by the speaker

(Radcliffe and Gogate, 1995) generates negative instantaneous acoustic intensity by

requiring that speaker volumetric flow rate track the negative of speaker face pressure.

qspkr = "KP face (1.2)
5

Acoustic Volume

3'. V) s. "9 as.
.»%§§K¢x,, .93;

 

     
     

Speaker

G AA S Tspkr

 

 

 

Figure 1.5. Schematic of the Active Acoustic Sink System

The AAS controller, OMS, is designed to realize the AAS objective (1.2). The

instantaneous acoustic intensity at the speaker face can then be expressed as

K
Aspkr

 

pfiace (1.3)

1,:-

For K > 0, acoustic power flux will always be directed into, and absorbed by, the speaker
face. As the magnitude of K is increased, more power flux will be directed into the
speaker faCe. The speaker behaves as active acoustic sink because any external noise
source located near the speaker will have its local acoustic intensity directed into the
speaker. The AAS can be ideally thought of as an acoustic black hole where acoustic
energy due to external noise at the face of AAS will not be reflected. The AAS system
generates negative, non-zero intensity, Ii, from non-zero acoustic pressure, p face. Active
cancellation systems use "anti-noise" to generate zero-valued face pressure resulting in
zero aeoustic intensity. This result indicates a fundamental difference between the
dissipative AAS approach and the non-dissipative "anti-noise" approach to active noise
control.

The AAS objective (1.2) requires speaker volumetric flow rate to track the speaker
face pressure with minimal magnitude and phase error over the controller bandwidth. This

is critical if AAS is to be successfully used for wide band noise control applications. A
6

1‘.

 

——-A ." . _. _

 

 

typical electromechanical speaker has a non-constant voltage-to-velocity frequency
response due to the speaker free-air resonance. This varying velocity response does not
allow the uncompensated speaker volumetric flow rate to accurately track a desired input
signal. Speaker compensation is therefore necessary to overcome this problem and allows
a compensated speaker to be used as a control actuator in AAS (Radcliffe and Gogate,
1996).

The successful performance of speaker velocity compensation depends on accurate
measurement of speaker cone velocity. The velocity measurement is done in this work
using a novel voice coil sensor and no external components. The design of the voice coil
sensor is based on an accurate speaker model which precisely predicts the measured
speaker dynamics (Radcliffe and Gogate 1995). It is the iterative process of model
development that is key to the successful AAS and velocity compensation development.
The identification of speaker dynamics is therefore essential in AAS.

The design stages of AAS include modeling and identifying speaker dynamics,
developing the speaker compensation design, and developing the control model and AAS
system. The final stage involves construction, implementation and testing of the active

acoustic sink controller.

1.4. Chapter Summary

Chapter 2 develops a dynamic model of an electromechanical speakers to be used
as control actuator in AAS. The data from speaker manufacturers do not always provide
the parameters required to construct a speaker dynamic model. A simple laboratory based
methodology for identifying the speaker parameters is also presented in this chapter. The
method is experimentally verified on an enclosed dual-wound coil subwoofer speaker.

Chapter 3 develops a velocity feedback compensation designed to minimize
magnitude and phase variations in speaker velocity response. Electromechanical speakers
have a non-constant velocity frequency response. This makes them poor control actuators

unless appropriate compensation is provided for their varying velocity response. Velocity
7

feedback compensation for a dual-wound coil subwoofer speaker is presented in this
chapter. It uses a proportional compensator and a novel speaker cone velocity sensor.

Chapter 4 develops an analytical model of AAS controller using a compensated
speaker model and an acoustic model of enclosed space. The AAS controller is designed
to give an acoustically absorptive surface at the cone of compensated speaker by
generating a negative, real acoustic impedance at the speaker face. A stability analysis is
performed to verify that the controller model results in a stable closed loop AAS system.

Chapter 5 discusses the implementation and testing of the AAS system.
Experimental tests of the AAS indicate successful performance.

Chapter 6 summarizes the dissertation and discusses directions for future research.

Chapter 2 Identification and Modeling of Speaker Dynamics

2.1. Introduction

Electromechanical speakers are commonly used as control actuators in many
acoustic control applications (Hull etal., 1990). Control of acoustic system behavior over
a specified bandwidth requires precise knowledge of speaker dynamics. This is essential
because often the speaker actuator bandwidth falls within the bandwidth of the acoustic
system for a realistic controller. The data from speaker manufacturers do not always
provide the parameters required to construct a speaker dynamic model. IEEE std. 219-
1975 gives methods to estimate the speaker parameters (Beranek, 1986, and Colloms,
1985) when partial information is provided by manufacturer, however, the standard
procedures are tedious. A simple laboratory based methodology for identifying the
speaker parameters of a dual-wound coil subwoofer speaker is presented in this chapter.
The methodology is validated by comparing the model and measured voltage-to-velocity

subwoofer responses. The effect of speaker enclosures is also discussed.

2.2. Modeling a Dual-Wound Coil Subwoofer Speaker

Dual-wound coil subwoofer (Figure 2.1) is common in the audio industry. It is an
electromechanical speaker (Figure 2.2) consisting of electrical, mechanical and acoustic
components. The dual-wound coil design consists of primary and secondary coils wound
on a bobbin connected to the speaker cone. The 12 inch (30 cm) subwoofer used in this
work (Radio Shack Realistic model 40-1350) is driven through the primary coil of the
speaker. As the drive voltage is applied to the primary coil, the varying electromagnetic

field produced around the primary coil interacts with the magnetic field produced by a
9

 

Figure 2.1. Radio Shack Realistic 12 inch, Dual-Wound, Subwoofer Model 40-1350

  
   
   

Air
Pemanent Magnet —> ‘ (acoustic)
(magnetic)
Primary Coil
1 ' 1 Second Coi Cone

Figure 2.2. Schematic Representation of a Dual-Wound Coil Subwoofer Speaker

fixed permanent magnet. The interaction between the two magnetic fields produces a
mechanical force in the primary coil attached to the speaker cone. The voltage in the
speaker cone's secondary coil is induced by both the velocity of the coil in the speaker

magnet's electromagnetic field and the mutual inductance between primary and secondary

coils.

An analytical model of the dual-wound coil subwoofer speaker can be developed

(Figure 2.3) by using the power summation and energy conservation principles of the

10

 

I.____..__ ........T............'...--....

Secondary CoilIRCOII

 

I I I s

t es_@ v5 IbIl 3

Sf' :bs(05 1| b I" I) 1?le ”S kr , I ‘
S:

L I r ~
e,”2 = As Is I I IS ‘
. I g Fspk . 1? P(t) g
r "W 1 Q0) .
21,, 1p I u- gpkrl ‘

l

 

I I 3

an. fl can u“ “ .- “.‘ " l.‘ .‘

E'Rectrical- - Magnetic Mechanical Acoustic

Figure 2.3. Bond Graph Model of a Dual-Wound Coil Subwoofer Speaker without

Enclosure

Bond Graph methodology (Rosenberg and Karnopp, 1983). The Bond Graph arrows

represent power flow from the electrical voltage, Se, applied at primary coil through the

model’s elements. Dual-wound coils are represented here as a multiport Icoil field in the

bond graph model. The electrical power is either dissipated, transformed or stored in the
model elements. Power is dissipated in resistive elements Rom-land Rspkr’ and
transformed both from electrical to mechanical power in the 'GY' element and from
mechanical to acoustic power in the 'TF' element. Power is stored as kinetic energy in the
'1' elements and as potential energy in the '0 elements. The open circuit on secondary coil

is represented by secondary coil current, is = 0 imposed by the source of flow, Sf.

The energy variables in the bond graph model include coil flux linkages,

zip and 115, speaker cone displacement, xspk, and speaker cone velocity, Vspkr- A linear

11

 

approximation is used to relate the flux linkages, 1p and 115, of multiport Icoil field to the

port currents i p and is in primary and secondary coils.

l =Ii (2.1)

M 7 i
an ’ins the inductance matrix, i=[,p
Is

I coil, p

where, I =
[McoiLs

J is the vector of port
Icoil,s

2
currents and 2. =(1lpj is the vector of coil flux linkages. 100,7, p and Icoil,s in the
S

inductance matrix are the self inductances of primary and secondary coils, whereas,
Mcoil,p and Mew-1’s are the mutual inductances between primary and secondary coils.
The self and mutual inductance terms in the inductance matrix are equal for audio dual-

wound speakers because the coils are designed to be identical in construction.

(2.2).

Mcoil,p= c0il,s = coil

(2.3)

Icoil,p = coil,s = coil

The speaker equations can be written from the bond graph in state space form by
choosing energy variables as state variables and appropriate input and output variables.
The input variables are speaker drive voltage, e(t), and acoustic pressure, P(t). The
output variables are speaker volumetric flow rate, Q(t) and speaker drive current in the

primary coil, ip(t).

 

 

 

 

 

 

- 0 1 O - '0 O “
xspkr _R (xspkr
_ v _ ‘1 spkr bl v + —SD (e(t)
dt spkr — Cspkrlspkr ISpkr (Ispkrlcoil) spkr Ispkr (PU) (2.4a)
1:2 VIP
0 *b1 ‘Rcoil O
_ Icoil _

12

(2.4b)

 

Q(t) 0 SD 0 xspkr
[i (U): 0 0 Vspkr
p Icoil A
P

The voltage, ebs(t), introduced in the secondary speaker coil is obtained from the
bond graph model (Figure 2.3) by summing the voltages across l-junction on secondary
coil side. These voltages include the voltage, ebs1(t), due to the mechanical motion of

speaker cone and the voltage, (3sz (t) due to the mutual inductance, Mm”, between the

coils.

ebs(t) = ebs1(t) +ebs2(t) (2.5)
The voltage, ebs1(t) , introduced due to mechanical motion of the speaker cone is linearly

proportional to the speaker cone velocity, vspk,

ebs1(t) = (bl)vspk,(t) (2.6)
The voltage, ebs2(t), in the secondary speaker coil can be written from the constitutive

relation (2.1) of the coils by noting that the secondary coil current, is = 0 due to the open

circuit on secondary coil.

. di (t)
mow». =Mcoufi (27)

Hence, the voltage, ebs(t), introduced in secondary speaker coil is

di
ebs(t) = (bl)vspk, 4.11400”??? (2.8)

Equations (2.4) and (2.8) define the model of a dual-wound coil subwoofer speaker. The
speaker parameters necessary to define the model are: mechanical inertia of speaker,
Ispkr’ mechanical compliance of speaker, Cspkr’ viscous friction of speaker, Rspkr,
electromagnetic coupling factor, (bl), speaker coil resistance, Rcoil’ speaker coil

inductance, [cm-1, mutual inductance, Mcoil’ and the equivalent speaker area , SD. With

the exception of mutual inductance, Mcoil , these electrical and mechanical parameters are

defined in IEEE standard 219-1975 for loudspeaker measurements.
13

2.3. Speaker Transfer Functions
The transfer functions of the subwoofer speaker system describe the dynamics

between mechanical and electrical subsystems of the speaker. One of these transfer

functions is the plant transfer function, Gspk,(s), from the primary coil drive voltage, e(t),
to the speaker cone velocity, Vspkr- The other transfer function describes the dynamics

between speaker cone velocity, vspkr’ and the voltage, ebs(t), introduced in the secondary

speaker coil . Denoted H1(s), this transfer function was the basis for speaker cone

velocity sensor design. The knowledge of both transfer functions is important to speaker

velocity feedback compensator design.

The plant transfer function, Gspk,(s), between drive voltage and speaker cone

velocity can be written from (2.4) by noting that volumetric flow rate, Q(t) is equal to

speaker cone velocity, vspkr’ times its effective area, S D-

(I’ll/Icon S

 

 

 

Gspkr(s) = 2 (2 9)
- I - R + bl . °
[spkr-93 + (Rspkr + Mil-5'2 + ( 1 + Rcorl spkr ( ) )s + Rem,
coil Cspkr I coil I coil Cspkr

The transfer function, H1(s), between speaker cone velocity, vspkr, and secondary
coil voltage, ebs , is obtained by taking the Laplace transform of (2.8)
M - I s
——Ebs(s) = (bl) + s———“"" P( ) (2.10)
Vspkr(s) Vspkr(s)

The transfer function between primary coil current, ip, and speaker cone velocity, vspkr,

can be written from (2.4) as
2 1
[p (S) = Ispkr"; + Rspkrs + /Cspkr
Vspkr( S) (blls

 

(2.11)

From (2.10) and (2.11)

= Ebs(s) =Mcoil 2 1 (buy
H1(s) Vspkr(s) (bl) (I,,,,,,s +Rspkrs+(/Cspkr+ Mcoil» (2.12)

Transfer functions (2.9) and (2.12) are the speaker transfer functions required for sensor

 

and speaker compensation.
14

2.4 Speaker Parameter Identification Procedure

Parameter identification for the speaker model is based upon the structure of the
model equations. Experimental tests to identify the speaker parameters neglect the coil
inductance, [cm-1 . Below 200 Hz, the electric time constant determined by coil inductance
and coil resistance is small compared to the mechanical time constant. This allows
neglecting the speaker coil inductance and reduces the speaker transfer function (2.9) to a

second order form

Vspkr(s) = (bl) / (Ispkr X Rcoil) S

 

(2.13)
E(s) 32 + ZCmns + (0,2,
2 1
where: (on = ——-— (2.14)
CspkrIspkr

l Cspkr (“)2
and C: — -— Rspkr + -—— (2. 15)

2 [spkr Rcoil

The speaker mechanical compliance, Cspkr’ can be independently measured from

speaker diaphragm displacement, xspkr, for a known force, F, using force and

displacement measurements.

)1: k
Cspk, = —%—' (2.16)

The speaker mechanical inertia , I spkr, is obtained from (2. 14).
I s l

Pkr = __—'7
Cspkr X 03n

where (0,, = 271T" (2.17)

The fundamental natural frequency, f", is the frequency at which the velocity response

phase passes through 0 degrees since the velocity frequency response starts with a 900

phase lead and is obtained from experimental speaker cone phase response (Figure 2.5).

The fundamental natural frequency, fn, is independent of the damping ratio, C . The

damping ratio, C , is obtained from the quality factor, Q.

15

__1_
-2Q

Quality factor, Q, is determined from the half power points on the experimental speaker

C (2.18)

cone velocity magnitude response (Figure 2.5) following the IEEE standard 219-1975 for

loudspeaker measurements.

fn
=—— 2.19
Q f2-f1 ( )

Frequencies f1 and f2 , are the frequencies at which the velocity of speaker falls to 0.707
(-3 dB) of its value at resonance. The electromagnetic coupling factor, (bl), can be

obtained from the experimental speaker cone velocity frequency response. Let the log-

magnitude of the response at frequency (01(rad/sec) be M. From (2.13), the log-

magnitude, M, of speaker cone velocity response at frequency 601 is given by

 

\/ [bl x 031mg — (1112)]2 + [2 x bl x Cmfmn ]2

M = zorog10
Ispkr X Rcoil X[((os21 - (012 )2 + (4c2me121 )]

(2.20)

 

Using (2.20), it is possible to determine the value of (bl). Viscous friction, Rspkr’ is

obtained from (2.15) and (2.18).

1 2
R, V 3”” -@- (2.21)

pk!“ = —
Q Cspkr Rcoil
The speaker coil resistance, Rm”, in (2.21) is measured with an ohmmeter. The
equivalent speaker cone area, SD, is taken as 87% of the speaker cone area computed
from nominal speaker diameter to account for the area taken by the flexible cone

connection to the speaker frame (Beranek, 1986).

16

2.5. Parameter Identification of a Dual-Wound Coil Subwoofer Speaker

The laboratory procedure described in previous section for identifying the speaker
parameters was validated using a dual-coil subwoofer speaker for which no parameters
were available. The velocity frequency response was measured by a Briiel & Kjaer Laser
Doppler Velocity-transducer Set Type 3544 (Figure 2.4). The Doppler Frequency shift in
the target beam caused by the vibration of the speaker cone was detected and converted
into a calibrated voltage output proportional to the cone velocity. A Hewlett Packard
Dynamic Signal Analyzer 35660A was used to obtain the frequency response of the
transfer function between speaker cone velocity and the input drive voltage to the
speaker. Fundamental natural frequency, f n = 21 Hz and quality factor, Q = 0.3944 were
obtained from the experimental speaker cone velocity frequency responses (Figure 2.5).

The computed speaker parameters from the experimental measurements are tabulated in

 

 

 

 

 

 

 

 

 

 

 

 

Table 1.1.
Laser Veloci -transducer
Set Type 35
HP Dynamic Signal Analyzer
_
—
—
_
_
Sourcel .’ . 1 Channel 2 H
Chan 1 1" l

 

 

 

  
 
 

 

:3:- = = I Retroreflective material
' ' Q Speaker

‘7 U
Amplifier unit\

Figure 2.4. Schematic of Experimental Setup for Measuring Speaker Velocity Response

17

Table 1.1 Identified Parameters for a Dual-Wound Subwoofer Speaker

 

 

 

 

 

 

 

I Parameter I Value
Mechanical Inertia, Ispk, 555 gm
Mechanical Compliance, Cspk, 1.03 mm/N
Viscous Friction, Rspk, 7.8 N-sec/m

Electromagnetic Coupling Factor, (bl) 5-4 N/ A
Coil Resistance, Rcoil 3.5 Ohms

Coil Inductance, [cm-1 2.8 mH
Equivalent Speaker Area , S 2 531 cm2

 

 

 

 

A analytical model (2.13) of the subwoofer speaker without speaker coil
inductance was then constructed and simulated in MatLab® using the parameters in
Table 1.1. The model and measured frequency responses are shown in Figure 2.5 and
have the same fundamental natural frequency, f": 21 Hz, indicated by phase response
zero crossings. At 300 Hz, the measured velocity response magnitude is 25 dB below its
value at 21 Hz resonance. This makes the effect of noise dominant above 300 Hz and
accordingly measurements above 300 Hz were found unreliable due to poor signal-to
noise ratio. The model and measured responses agree well up to approximately 200 Hz.
After 200 Hz, the difference between model and measured frequency responses is

attributed to the effect of speaker coil inductance neglected in the speaker model (2.13).

18

 

ii -5. I. .I.! .. .ldl:\r.‘-rt.j . ..

 

 

 

Log-Magnitude (dB)

 

 

 

 

Phase (Degrees) N

 

 

 

4 10 Frequency (Hz) 100 300
Figure 2.5. Comparison of Measured and Speaker Model Velocity Responses without
Speaker Coil Inductance (dB re 1)

The analytical model (2.9) of the subwoofer model with speaker coil inductance
was constructed and simulated in MatLab®~ A speaker coil inductance of 2.8 mH was
used to obtain a good fit between the model and measured responses (Figure 2.6). The
close agreement between the model and measured responses indicate that the presence of

speaker coil inductance in the speaker model is essential.

19

 

.l D .Ui..

 

 

a

 

 

 

 

 

 

 

 

 

100 300

Frequency (Hz)

Figure 2.6. Comparison of Measured and Speaker Model Velocity Responses with
Speaker Coil Inductance (dB re 1)

2.6. Speaker Enclosure Effects

The enclosure which supports the speaker drive unit has an acoustical effect of

increasing the speaker free air resonance. This effect is represented by acoustic

compliance, Cbox’ and acoustic resistance, Rbox- in the bond graph model of speaker
(Figure 2.7). The acoustic pressure inside the speaker enclosure is denoted by Pint“)
while P(t) represents the speaker face acoustic pressure. The speaker drive voltage, e( t),
and the speaker face pressure, P(t) form the two inputs for the speaker system. The

acoustic compliance, Cbox’ models the flexibility of air inside the enclosure and is given

by (Beranek, 1986)

20

V

 

bl I

ebS (t ebs I: ®Dvspkr . u I Rbox Cbox
S f |———)>l I I—7 GY I spkr RSpkr : «Dim»:

s=0 , . I r
3:252:22: 13 I I / SD Pint(t)1Q(t)

ISzmondary EOil-Rl-b c2271.- _ - r “l “I “““““ I lSpeaker Enclosure.
' 1

I

'l

I la

 

 

 

 

. F . .
I(3011 I 1 I__Aspkr TF |__- 1 ‘
. I Vspkr I P .
A p 1p 1;! ;I I (I) Q“) I
e( t) - -

e_z"7:')>'1|——> GY 'Cspkr ' , l
p L I I [Speaker Face] :
| I I I
ILPrimary Coil] Rcoil I g I I

Electrical Magnetic Mechanical Acoustic

Figure 2.7. Bond Graph Model of Dual-Wound Coil Subwoofer Speaker with Enclosure

 

V .
Ca. = b”; (2.22)
paircair

where, Vbox is the volume of speaker enclosure. The volume of the speaker is subtracted
from the actual volume of enclosure in order to obtain Vbox- As a first approximation, the
volume of speaker in meters3 equals 0.4 times the fourth power of the advertised diameter
in meters (Beranek, 1986). The acoustic resistance, Rbox’ models the pressure drop
associated with the leakage in the speaker enclosure.

The state-space equations for the speaker with enclosure can be written by

selecting energy variables : speaker cone displacement, xspkr, speaker cone velocity,

vspkr, primary coil flux linkage, 2p, and volume displacement, Vbox as state variables.

The input variables for the state-space model are speaker drive, e(t), and speaker face

pressure, P(t) while the output variables are speaker volumetric flow rate, Q( t) and
primary coil current, ip( t).
21

 

 

 

 

 

 

 

 

 

 

 

 

 

i 0 1 o o ' '0 o _
_ —R _
(xi-pk.) C i I spkr (I b: ) CSD (xspkr‘ 0 —SD
k k k ' _
i Vspkr - sp r sp r sp r spkr c011 and Vspkr + [spkr (80))
(It a, 0 _b1 -R.,,-, O 1,, 1 0 (Po)
\ Vbox ) [coil \Vbox )
0 SD 0 ——’1—— _o o _
_ Rbobeoxd
(2.233)
1" “(xspkr\
0 SD 0 o
Q(t) = Vspkr (2 23b)
ipm 1 11 .
O O O p
L Icoil -\Vbox )

This state-space equation is similar to (2.4) with an additional state, Vbox’ corresponding

to the volume of the speaker enclosure.

Enclosure effect was verified on the dual-wound coil subwoofer speaker for which
parameters are given in Table 1.1. The speaker was placed in a rectangular enclosure of
dimensions 35 cm x 24 cm x 55 cm. From (2.22), the acoustic compliance of the
enclosure is Cenc, = 3.25 x10—7 m3/Pa. The analytical model of the enclosed speaker
from input voltage, e(t) to speaker cone velocity, vspk, was obtained from (2.23) and
simulated in MatLab®. An acoustic resistance of 2.78 x104 Pa-sec/m3 was used to
obtain a good fit between the model and measured responses (Figure 2.8). The measured
and model responses have the same fundamental natural frequency, f" = 65 Hz, shown
by phase response zero crossings. The enclosure thus has the effect of increasing speaker
free-air resonance. At frequencies above 300 Hz, the model response diverges from the
measured response. This effect can be attributed to the standing wave acoustic modes of
the speaker enclosure. The first acoustic mode along 55 cm direction of the speaker
enclosure occurs at 313 Hz. It has an influence on the magnitude and phase responses at

frequencies below 313 Hz and affects the low-frequency speaker behavior.
22

 

 

 

 

 

Log-Magnitude (dB)

 

 

N

00

 

 

 

Phase (degrees)
C:

 

 

g A - - x l

“8’

 

100 i 300

A

Frequency (Hz)

Figure 2.8. Comparison of Measured and Enclosed Speaker Model Velocity Response (dB
re 1) i

The velocity frequency responses shown in Figure 2.8 have limited bandwidth and
large phase shifts that prevents speaker cone velocity tracking of desired velocity inputs.
This makes it an unsuitable actuator in AAS where speaker cone velocity must track the
speaker face pressure with minimum magnitude and phase variation. These bandwidth
limited speakers are unsuitable acoustic control actuators unless compensation is provided

for their varying velocity response.

2.7. Summary
A simple laboratory based methodology for identifying speaker dynamics for

acoustic control applications has been presented and experimentally validated in

23

 

9Fa'1 '

 

laboratory tests on a dual-wound coil subwoofer speaker. The analytical model response
was found to agree well with the measured response. The effects of speaker enclosure was
also discussed. The enclosure was shown to increases the speaker free air resonance.

The speaker velocity response has limited bandwidth and large phase variations.
These magnitude and phase variations make speakers ineffective control actuators for
acoustic system where input is proportional to actuator input velocity. Compensation for
the varying velocity response of speakers is needed to make them effective acoustic

control actuators.

24

Chapter 3 Velocity Feedback Compensation of Speakers

3.1. Introduction

Electromechanical speakers commonly used as control actuators in acoustic
applications have a non-constant voltage-to—velocity frequency response due to the free-air
resonance of the speaker (Figures 2.6 and 2.8). Speaker volumetric flow rate cannot
accurately follow the controller output signals if the variations in velocity frequency
response occur within the acoustic control system bandwidth. This makes
electromechanical speakers ineffective as control actuators unless a compensation is
provided for their varying velocity response.

The idea of compensating speaker velocity response dates back to early 19205.
Articles on this subject (Harwood, 1974, Klaassen, et al., 1968, Holdaway, 1963, Werner,
1958, Holle, 1952, Tanner, 1951) mention the use of velocity feedback as the
compensation technique. They, however, differ on the method used for sensing speaker
cone velocity. The earliest mention of speaker compensation is found in a patent (no.
231972) awarded to P. G. A. H. Voigt on Jan. 29th 1924. In his patent, Voigt used back
emf induced in the speaker coil by its motion in the magnetic field as a measure of speaker
cone velocity. The difficulties reported in his implementation (Harwood, 1974) were the
needs to compensate for temperature induced changes in speaker coil resistance and
frequency dependent variations in speaker coil inductance. Another speaker compensation
is found in a patent (no. 272622) by A. F. Sykes dated March 20‘h 1926. He used voltage
introduced in an auxiliary coil to sense the speaker cone velocity. The method was not

successful because the mutual inductance between auxiliary and speaker coil introduced

25

errors in the speaker cone velocity sensing. In a third implementation, M. Trouton in his
patent (no. 320713, Aug. 10th 1928) used the voltage obtained from speaker cone
displacement in a capacitive method to sense speaker cone velocity. The method though
relatively simple, required a considerable spacing between capacitive elements for large
amplitudes motions of speaker cone. Non-axial movement of the speaker coil also
introduced error in this method. The use of accelerometers to measure speaker cone
velocity has also been mentioned (Klaassen et al., 1968). In this method, the output of
accelerometer attached to the speaker cone diaphragm is integrated to obtain voltage
proportional to speaker cone velocity. The obvious disadvantages in this method are the
inaccuracies in integrator due to noise and accelerometer mass loading of the speaker
cone. The novel speaker cone velocity sensor developed in this work uses the mutual
inductance effect compensated voltage introduced in an auxiliary speaker coil. The
compensated auxiliary coil voltage provides an accurate velocity sensor and does not have

the problems associated with sensors mentioned above.

3.2. Speaker Velocity Feedback Design

Velocity feedback compensation (Figure 3.1) for minimizing the magnitude and
phase variations in speaker velocity response is designed and demonstrated in laboratory
tests on a dual-wound coil subwoofer speaker. The feedback compensation uses
proportional controller, K p to generate drive voltage, e(t) at the primary set of speaker
coils. The speaker cone velocity is obtained through sensor transfer function, H (s). The
closed loop system transfer function, T(s) , from block diagram (Figure 3.1) is

Vspkr“) = Kstpkr (s)

(3.1)
Vd(s) 1+ Kstpk,(s)H(s)

 

Tspkr (S) =

where, Vd(s) is the Laplace transform of desired velocity input, vd (t). Gspk,(s) is the

open-loop transfer function of enclosed subwoofer speaker. As the proportional

compensator gain, K , is increased, the closed loop transfer function approaches 1/ H ( s).

If the sensor transfer function is a real constant, B , over the controller bandwidth, (1),, ,
26

 

 

 

 

Vd (t) Proportional e(t) s aker vspkr(t)
Controller Disamks —>
— Gspkr
Velocity

Sensor
H ( s)

 

Figure 3.1. Schematic of Speaker Velocity Feedback Compensation

H(ja))=fi V we[0,wb] (3.2)
the closed loop transfer function, Tspk, (s), will approach a constant, 1/ [3 , with zero phase.
This compensation forces speaker cone velocity to accurately follow the desired velocity
input. The result is independent of the speaker dynamics provided the sensor has a
constant transfer function (3.2) over the controller bandwidth. Proper selection and design

of the velocity sensor is critical.

3.3. Speaker Cone Velocity Sensor Design

An accurate speaker cone velocity sensor is necessary for the successful
performance of closed loop speaker velocity feedback system. A novel cone velocity
sensor design for a dual-wound coil subwoofer speaker is discussed here. It uses speaker
cone velocity induced voltage in a secondary speaker coil obtained by compensating the
mutual inductance effect between dual-wound coils.

The transfer function from speaker cone velocity to secondary speaker coil voltage
for the enclosed dual-wound coil subwoofer can be obtained from the state-space equation

for the enclosed speaker. The voltage, ebs( t), introduced in secondary speaker coil is

given by (2.8).

di
ebs(t) = (bl )vspk, + Ma,” 71:3- (3.3)

27

The transfer function, H1(s), between speaker cone velocity and secondary coil voltage is

obtained by taking Laplace transform of (3.3)

Em“) =(bl)+SMCOile(S)

(3.4)
Vspkr (‘9 ) Vspkr (s)

The right hand side of (3.4) can be manipulated using the following Laplace domain

equation obtained from (2.22).

 

SD
V = V 3.5
b0x(S) S +(1/Rb0be0x) Spkr(s) ( )
R 1 bl S x
(s2 + iii-’51s + —)V,p,,,(s) = Lilpm - J—vaoxm (3.6)
spkr Cspkr spkr Ispkr Cboxlspkr

From Equations (3.5) and (3.6)

 

2
(s2 + —————Rs”k' s + ——1 )Vspkr( S) = (bl)s 1 JSDVSPkr(s) (3.7)

— I p ( s ) — [I -
Ispkr CspkrIspkr Ispkr (Rbobeox )3 +1 Cboxlspkr
From (3.4) and (3.7), the transfer function, H1(s), between speaker cone velocity and

secondary coil voltage can be written as

2
Ebs“) _ Meal/133 + 33 + Cs + D]
Vspkr“) (bl) X (Rbobeoxs +1)

 

H1(s) = (3.8)

Where, A = (Rbox I@erwa ); B = (Ispkr + Rbox Rsperbox)

R C 2 (bl)2C M - mm2
C = [ Rspkr + ng box + SD Rbox + Rbox box } D =[ [gall C
spkr coil coil spkr

 

 

One of the zeros of the transfer function H1( s) can be approximated by (Rbobeoxs +1).
This reduces [11(3) to a complex second order zero. The validity of this assumption is

verified later in computer simulations.

 

 

M - 1 (bl)2
H = CO" I 2 + R + + 3.9
1(3) (bl) |: Spkrs spkr-9 [ Ceq Mcoil]:| ( l

28

where

__ Cbox Cspkr
eq — 2
Cbox + SDCspkr

 

(3.10)

is defined as the equivalent speaker compliance due to speaker mechanical compliance,

Cspkr, and the enclosure acoustic compliance, Cbox-

The measured and analytical frequency responses of the transfer functions (3.8)
and (3.9) for the dual-wound coil subwoofer speaker (Table 1.1) placed in an enclosure of
dimensions 35 cm X 24 cm x 55 cm are shown in Figure 3.2. From section 2.6, the
enclosure acoustic compliance, Cenc, =3.25x10_7 m3/Pa. This gives an equivalent
speaker compliance of 0.104mm/N. The analytical frequency responses are obtained by
simulating the transfer functions (3.8) and (3.9) with an equivalent speaker compliance of
0.104 mm/N while other speaker parameters are obtained from Table 1.1. The mutual
inductance, Mcoil’ was not supplied by the speaker manufacturer and a value of 1.2 mH
provided a good fit between model and measured responses. A Hewlett Packard Dynamic
Signal Analyzer 35660A was used to measure the frequency response between speaker
cone velocity and secondary speaker coil voltage. The measured frequency response
agrees well with the analytical model responses over 4-300 Hz bandwidth. The close
agreement between the second (3.9) and third order model (3.8) responses justifies the
pole-zero cancellation of (Rbobeoxs +1) in (3.8). It allows the transfer function, H1(s),
to be represented by a damped second order zero (1; = 0.24) at 122 Hz (3.9) as indicated
by 90° phase crossing (Figure 3.2).

The secondary speaker coil voltage due to mechanical motion of speaker cone is
linearly proportional (2.6) to the speaker cone velocity through electromagnetic coupling
factor, (bl). It provides a good measure of speaker cone velocity below 70 Hz as seen
from Figure 3.2. Over this frequency range, the phase is nearly 0° and the magnitude
varies from 13.5 dB to 14.8 dB which is the electromagnetic coupling factor, (bl). At

frequencies above 70 Hz, the second order zero at 122 Hz due to mutual inductance

29

 

N42.
00

 

 

—' Second Order Mode

 

Log-Magnitude (dB)
c

 

 

 

 

-20 _ ..................................
Third Order Model
4 10 Frequency (Hz) 100 300
200

 

 

 

_ Second Order Model

Phase (Degrees)
o

 

 

 

 

Third Order Model
100 300

 

11:
8
is
522222122222: IIIIIIIQI'I

Frequency (Hz)

Figure 3.2. Secondary Coil Voltage/Speaker Cone Velocity Model vs. Measured
Responses (dB re 1) A

between the dual-wound coils causes large magnitude and phase changes. The secondary
speaker coil voltage becomes an inaccurate velocity sensor above 70 Hz unless the sensor
is compensated for the second order zero at 122 Hz. A second order filter, HC (3) , with an
undamped natural frequency of 122 Hz and a damping ratio of § = 0.24 was built to
compensate for this mutual inductance zero.

The mutual inductance compensated frequency response between speaker cone
velocity and secondary speaker coil voltage has less than 5 dB gain variation over 4-
300 Hz (Figure 3.3) frequency range compared to 30 dB variation for uncompensated

secondary coil voltage (Figure 3.2). Over the same range of frequencies, phase varies less

30

than 20° for filtered secondary coil voltage compared to a variation of 160° for
uncompensated secondary coil voltage. Although some magnitude and phase variations
remain, the filtered secondary speaker coil voltage provides a more accurate measure of
speaker cone velocity. It provides a speaker cone velocity sensor whose transfer function,
H ( s ), can be assumed to be unity with zero phase over 4-300 Hz frequency range.

This velocity sensor does not have the drawbacks of previously mentioned sensors.
There is no need to account for the temperature induced variations in speaker coil
resistance and frequency dependent variations in speaker coil inductance as in Voigt's

method. The sensor does not mechanically load the speaker cone as in the velocity

 

 

 

 

 

 

 

 

 

 

 

 

 

40 r I I II '
£5, 20 ------------ 2 ------- -------
3 ' ' ' E E E E E 55“ i
go 01 .............. -. ............ g j A; 17': —“""M
a i E i i i §§
2' -20 _ ............................ 3 ....... 5 ..... ........................
on E E E E E SE
3 40 - . g .= s a 3 3' .
4 10 Frequency (Hz) 100 300
200 r r
’e‘
2
00
5’07 e - Am ........ T. ...... -
D
E
a.
_200 Z] ' Iii I
4 10 Frequency (Hz) 100 300

Figure 3.3. Compensated Secondary Speaker Coil Voltage/Laser Velocity Response (dB
re 1)

31

sensors using accelerometers. There are no geometric inaccuracies associated with using
capacitive pick-up elements. This unique velocity sensing mechanism requires only the
filter and does not need any changes in the physical parameters of the subwoofer speaker.

It yields a viable velocity sensor for the feedback compensation of the speaker system.

3.4 Stability Analysis of Speaker Velocity Feedback System
A closed loop stability analysis of the speaker velocity feedback system determines

the maximum stable gain of the proportional controller (Figure 3.1). The closed loop

system transfer function, Tspk,(s), is given by (3.1) where, K stpk,( s)H ( s) is the open-

loop transfer function. The maximum stable gain, K “max of the proportional controller is

given by the gain margin, GM, of system open-loop transfer function.

1
Kstpkr(jwl )H(jml)

 

KM,“ =GM= (3.11)

where, (01, the phase crossover frequency, is the frequency at which the phase angle of the
open-loop transfer function becomes —1800. The gain margin can be evaluated from the
bode diagram which is a plot of magnitude and phase of open-loop transfer function
versus frequency.

The bode plot of the speaker velocity feedback system (Figure 3.1) for K p =1 was
experimentally measured (Figure 3.4) over 4-300 Hz frequency range. The measured
phase response does not cross -l80° over 4-300 Hz indicating that the closed loop speaker
system will be stable with infinite gain margin. It must be noted that the phase response
will eventually cross -180° at some frequency above 300 Hz due to the minimum phase
nature of the open-loop transfer function of the enclosed speaker. This will reduce the
gain margin and thereby limit the gain of the proportional controller. The 20 dB/decade
roll off in the magnitude response, however, will provide enough gain margin at the phase
crossover frequency above 300 Hz before destabilizing the closed-loop system. This fact

was verified by plotting an analytical root locus plot of the speaker system.

32

 

 

 

Log-Magnitude (dB)

 

 

 

 

 

 

 

 

 

 

 

200 : : l I T I I ' T

é or“:v- ................................... -

D

g .
a.

-2004 ' 1.110 2 i I i I C 2:160 I 300

Frequency (Hz)

Figure 3.3. Measured Bode Plot of the Open-Loop Transfer Function of Speaker Velocity
Feedback System (dB re 1)

A root locus of the closed-loop speaker velocity feedback system (Figure 3.1)
using an analytical model of enclosed speaker (2.22) is shown in Figure 3.5. The velocity

sensor transfer function is assumed to be unity with zero phase (Figure 3.3). The root

locus is plotted by varying the proportional controller gain, K p, from 0 to 110 and

indicates the location of closed-loop poles in complex plane as a function of K p. It is

clear from the root locus that closed-loop poles lie in the left-half complex plane
indicating a stable closed-loop system. The limiting value of the proportional controller
gain at which one of the closed-loop poles becomes zero is found from the Routh-Hurwitz

criteria to be 105. The closed-loop system thus becomes marginally stable at a

33

2000

 

 

 

 

 

 

 

 

 

 

OjenLoopPoles:
1500 -1.07><103 —
-2.38x102:4.11x102j
1°00 -1.11x101 ‘
OpenLoopZeros:
E 500_ ............................................ 2 _
:é -l.23x10
E‘ 0
.a 0 ?
a
E
_500_ .......................................................................................... ..
4000.. .......... ............ ............ .......... _
4500,. .......... ............ ............ .......... -

 

 

 

 

 

493800 -1500 -1000 1500 0 500 1000 1500 2000
Realeis ~

Figure 3.5. Root Locus of Velocity Feedback System for Varying Proportional Controller

Gain

proportional gain of 105. This indicates an almost infinite gain margin and this was
expected from the experimental results shown earlier. It is thus possible to implement the
velocity feedback controller over a low frequency bandwidth without destabilizing the

closed-loop system.

3.5. Performance Evaluation of Speaker Velocity Feedback

The performance of the speaker velocity feedback compensator was evaluated by
measuring (Figure 3.6) the closed loop transfer function, Tspk,( s), of the speaker velocity
34

feedback loop (Figure 3.1) for different proportional compensator gains (Figures 3.7-
3.10). The speaker cone velocity was independently measured using laser velocimeter.
As the proportional compensator gain is increased, the magnitude of the closed loop
speaker velocity response becomes more constant and phase change is minimized. At a
compensator gain of K p = 200, the magnitude varies less than 5 dB from 4 Hz to 300 Hz
as compared to an open loop magnitude change of 20 dB. Over this range of frequencies,
the phase angle changes less than 20 degrees as compared to an open loop change of 160
degrees. At high proportional gains, the closed loop transfer function (Figure 3.10)
approaches the inverse (3.2) of the sensor transfer function, H (s) as expected (3.1). When
compared to the uncompensated speaker, the feedback compensated subwoofer speaker

has much lower magnitude and phase variations in the velocity response.

    
  

Radio Shack Realistic 12 inch,
Dual—Wound Enclosed Subwoofer
(Model No. 40-1350)

. '. “~ Electronics for Speaker
Laser Velocrty-Transducer - . g 8' Velocity Sensor and
Set Type 3544 i .. .. . ‘0 ' Velocity Feedback

Figure 3.6. Experimental Setup for Speaker Velocity Feedback

35

 

 

3

 

N
C

 

ti)
o 4

 

 

 

 

 

Log-Magnitude (dB)
O

s

 

 

4 Frequency (Hz) 100 300 4 Frequency (Hz) 100 300

Figure 3.7. Comparison of Measured and Model Closed-Loop Speaker Velocity Response
for Proportional Gain of Kp =10 (dB re 1)

 

 

 

 

A 200 u
% 7,; =
v K 50
o 8 P
“U Sb 2
.3. 5
So 2:
2. 2
3° 0.
~200

 

 

 

 

 

 

300 4 Frequency (Hz) 100 300

Figure 3.8. Comparison of Measured and Model Closed-Loop Speaker Velocity Response
for Proportional Gain of K p = 50 (dB re 1)

 

 

.b

O
N
8

N
C

 

 

 

Phase (Degrees)
o

 

 

 

 

Log-Magnitude (dB)
o

is
O

r
1'9
8

 

 

Frequency (Hz) 100 300 4 Frequency (Hz) 100 300

Figure 3.9. Comparison of Measured and Model Closed-Loop Speaker Velocity Response
for Proportional Gain of K p = 100 (dB re 1)

36

 

 

.b
O
N
8

 

 

N
O

...............................

Log-Magnitude (dB)
2%
-§ 1 E
Phase (Degrees)
o

...........................................

 

 

 

 

L

00 300

L
O
to
8

 

 

100 300‘

.5

Frequency (Hz) Frequency (Hz) 1

Figure 3.10. Comparison of Measured and Model Closed-Loop Speaker Velocity
Response for Proportional Gain of K p = 200 (dB re 1)

The feedback compensation uses simple control technology, is easy to implement
and the results are especially significant because no changes in speaker physical
parameters are required. Compensation of speaker response allows speaker cone velocity
to accurately track desired velocity inputs such as the output of an acoustic controller. It

makes limited bandwidth subwoofer speakers effective acoustic control actuators.

3.6. Summary

Velocity feedback compensation for minimizing the magnitude and phase
variations in the velocity frequency response of a dual-wound coil subwoofer speaker is
presented in this chapter. The feedback compensation uses proportional controller to drive
the subwoofer through primary speaker coils. The speaker cone velocity sensing is done
by a novel velocity sensor designed using the bond graph model of subwoofer. It uses
speaker cone motion induced secondary coil voltage obtained by compensating the mutual
inductance effect between dual-wound coils.

Speaker velocity feedback compensation is experimentally shown to reduce
speaker velocity magnitude and phase variations. The compensated speaker has variations
less than 5 dB in gain and 20 degrees in phase over 4400 Hz bandwidth compared to 20

dB and 160 degrees variations for the uncompensated speaker. This allows compensated
37

subwoofer speaker to accurately follow any desired velocity input and makes it an

effective acoustic control actuators in AAS.

38

Chapter 4 Active Acoustic Sink Control Model Development

4.1. Introduction

An analytical model of an AAS controller is developed in this chapter using a
compensated speaker model and an acoustic model of enclosed space. The AAS uses
velocity compensated speaker to absorb acoustic energy. The speaker compensation is
shown to be necessary for the successful performance of AAS. A theoretical analysis is
done to show that a negative, real acoustic impedance at the speaker face results in local
acoustic power flux into the speaker. The AAS controller is designed to give an
acoustically absorptive speaker surface by generating a negative real acoustic impedance
at the speaker face. A stability analysis is performed to verify that the controller model
results in a stable closed loop AAS system. The resulting controller model provides
maximum possible absorption of local acoustic power flux over a wide range of noise

frequencies without destabilizing the AAS system.

4.2. Theory of Active Acoustic Sink System

Theoretical analysis shows that a negative, real acoustic impedance at the speaker
face results in local acoustic intensity directed into the speaker. An Intensity analysis
based on an acoustic energy balance equation is applied here to a control volume
containing an electromechanical speaker. It accounts for the rate of change of acoustic
energy within a volume, rate of acoustic energy outflow from the volume and power

radiated by any sources within the control volume.

39

 

Consider a closed, piecewise smooth surface, S, bounding the control volume, V
which encloses an electromechanical speaker (Figure 4.1). An area element of surface S

is denoted by dS with an outward unit normal vector, ii. The speaker with a source
qspkr

S

strength of is idealized as a monopole flow source located at x0 in the control

volume.

qspkr _ Aspkrvspkr
Vs Vs

 

(4.1)

where, Aspk, is the effective area of speaker cone and vspk, is the speaker cone velocity.
V, is the volume of a pulsating sphere whose flow rate is equivalent to the volumetric

flow rate introduced by the speaker cone motion.
The principles of mass and momentum conservation applied to the control volume

result in following vector equations

_1_.3_P=_pov.v+po ‘13P” 6(x—x0) (4.2)

 

c2 at Vs

 

 

Figure 4.1. Control Volume containing Idealized Point Flow Source

40

 

.3 fair:
.

I!" H: MP“... .

‘

av ..
p057+vp=0 (4.3)

where, c is the speed of sound in air, p0 is the equilibrium density of air, p is the

fluctuation in pressure from its equilibrium value and i3 is the vector of particle velocity
fluctuations. 6(x —x0 ) is the three-dimensional delta function and has properties
analogous to the one-dimensional Dirac delta function.

The acoustic energy balance equation can be written by adding the equations

obtained by multiplying (4.2) with p/po and taking the dot product of i3 with (4.3).

 

3E " - (Ispkr
—— + V o I = 5 x — x ' 4.4
l _. _. p2 - .. .
where E = 5pc v o v + 7 is the acoustic energy density and I = pv is the acoustic
c

intensity. The first term on the left side of (4.4) represents the rate of change of energy in
the control volume while the second term represents the rate of outflow of energy from the
control volume. The term on the right side of (4.4) represents a source imparting acoustic
energy to the control volume through a "monopole" speaker. Equation (4.4) can thus be
interpreted as the acoustic energy conservation law applied to the control volume, V and

forms the basis for further analysis. Taking the time-average of (4.4), we get

T—)°o T-—>°°

T T
. E(T)-E(0) . _1_ '° . -° = . _1_ ‘13 kr _
lzm {—T )+ hm r g (V I)dt T132” g (p—‘fs—aa xo))dt (4.5)

Noting that the acoustic energy density, E(T), inside a control volume is bounded as

T -—) ca, (4.5) reduces to

Nil

.. 1 _
V . = quspkr6(x -x0) (4.6)
S

where, 7 and pqspk, denote the time-averages of 7 and pqspk, respectively.

The acoustic power output, W , from the control volume is given by

41

W=jioiids (4.7)
S

Using (4.6) and Gauss's divergence theorem, (4.7) can be written as
.. _. 1 __
W = I ‘j/ j V . IdV =7];- [‘1’] pqspkr5(x —x0 )dV (4.8)

The three-dimensional delta function exhibits the "sifting" property for a continuous

function, f (x) in three—dimensional space

f(y), y within V

Iyflxwfl - ”(IV ={ O, y outside V (49)

Assuming that the time-averaged quantity, pqspk, is continuous within the control

volume, the relation (4.8) for acoustic power, W , can be written as

1 __
W = Vp(xo )QSpkr (4°10)
s

If the speaker volumetric flow rate, qspkr’ is chosen such that it tracks the acoustic

pressure, p(xo ), at the speaker face,

qspk,(x0 ) = -Kp(x0 ) ‘ (4.11)
a negative real speaker face acoustic impedance will result. With the choice of speaker

volumetric flow rate in (4.11), the acoustic power output from the control volume is

 

W =—-I£p2(x0) (4.12)
VS

I For K > 0, it is clear from (4.12) that the acoustic power in the control volume will always
be negative. This corresponds to power absorbed by the control volume. Any external
noise source located near the control volume will have its local acoustic energy directed
into the control volume causing it to behave as an active acoustic sink. The
electromechanical speaker can thus be made to act as an active acoustic absorber if the
AAS controller is designed to realize the control law (4.11). It has been postulated

recently (Madanshetty, et al., 1994) that other choices of speaker volumetric flow rate,

42

qspkr’ will result in acoustic power absorption. The derivation above clearly indicates

choice of speaker volumetric flow rate in (4.11) is required for acoustic power absorption
by the speaker.

The volumetric flow rate of absorptive speaker must track the acoustic pressure at
the speaker face as per the control law (4.11). It is critical that this tracking occurs over
the controller bandwidth with minimum magnitude and phase error if AAS is to be
successfully used for wide band noise control applications. Since the speaker volumetric
flow rate is linearly related to the speaker cone velocity according to (4.1), speaker
velocity must track the speaker face pressure for (4.11) to be satisfied. It was shown
earlier that the cone velocity of a conventional electromechanical speaker cannot
accurately follow the desired signal (Figures 2.5-2.6, and Figure 2.8) due to the speaker
free air-resonance. This makes conventional electromechanical speakers inappropriate
acoustic absorbers unless a compensation is provided for their varying velocity response.
A practical method of speaker compensation using velocity feedback was discussed in

Chapter 3. The compensated speaker is used as effective control actuator in the AAS.

4.3. Active Acoustic Sink Controller Design
The AAS controller is designed to yield an acoustically absorptive speaker cone
surface resulting in local acoustic intensity being directed into the compensated speaker.

The block diagram of the AAS system is shown in Figure 4.2. A microphone with a
sensitivity of Smic is placed at the speaker face and measures the total acoustic pressure,

p face at the speaker face consisting of the pressure components due to the motion of

speaker cone surface, pspkr’ the pressure component due to incident acoustic waves from
ambient noise, p,- and the pressure component due to reflected acoustic waves from the
speaker face, pr.

pface = pspkr + pi + pr = pspkr +(1+ 9“(0))“ (4-13)

where, 9i(a)) is the pressure-amplitude reflection coefficient defined as the ratio of

43

Compensated Room
Speaker Acoustics

Vspkr _‘ qSPkr pspkr
_ an»

 

 

 

 

 

 

 

 

 

 

Figure 4.2. Block Diagram of Active Acoustic Sink

Fourier transform of reflected wave pressure, p,, to the incident wave pressure, p,-

(Pierce, 1981). The signal, ep( t), from the microphone is an input to the AAS controller.

The output of the AAS controller transfer function, GAAS’ is the desired signal, Vdes , for

the compensated speaker. The volumetric flow rate of the compensated speaker, QSpkr’

acts through the room acoustics transfer function, Groom, to generate the acoustic pressure

component, Pspkr- This room transfer function generates in a closed loop process. The

relation between speaker volumetric flow rate, qspkr’ and the speaker face pressure, p face
is obtained from the block diagram of the closed loop AAS system.
qspkr = (SmicGAAS TspkrAspkr )P face (4.14)
The required AAS controller transfer function, GAAS’ is obtained by substituting

the AAS design objective (4.11) into the system model (4.14) and rearranging terms

—K
SmicAspkrTspkr (4. 1 5)

 

GAAS =

Since the speaker transfer function, Tspk,(s), approaches a real constant, 1/13, at high

levels of compensation (Figure 3.10), the AAS controller transfer function simplifies to a
pure gain.
0M5 = i (4.16)

SmicAspkr
44

The only non-static terms in (4.16) is the microphone sensitivity, Smic. The radio shack

microphone used in the implementation of AAS controller has a nearly constant transfer

function in 50-1000 Hz frequency range.

4.4. Stability Analysis of Active Acoustic Sink System

A stability analysis is needed to determine the maximum stable, closed loop
controller gain, K, in the presence of room acoustics, Groom, and compensated speaker

dynamics, Tspkr- The closed-loop transfer function, Tcl(s), of the AAS system (Figure

4.2) with controller (4.16) is

 

P -K T G
TC,(s)=M=(1+9t)—C;2L(i)—=(1+9t) l3 ‘Pk’m mm“) (4.17)
Pi(s) l-G0,(s) l+KfiTspk,(s)Gmom(s)
where, 001(3) is the open-loop transfer function
G0,(s) = —K[3Tspk,(s)Gmom (s) (4.18)

The closed-loop system transfer function (4.17) will be unstable if 1— Go,(ja)) = O, for

any a). In other words, if the open-loop transfer function is such that
KBTspk,(s)Gmom(s) = —1 for s = ija), Va) 2 O - (4.19)

the closed-loop AAS system will be unstable. This can be stated in terms of the usual
Nyquist criteria for negative feedback systems by considering

G010) = KBTSpk,(s)Gmom (s) to be a modified open-loop transfer function of the AAS

system.

Nyquist Criteria : Let the curve C2 in complex plane be the mapping of a closed curve,
C1 completely enclosing the right half of the s - plane through a stable transfer function,
G010). If G0,“) is analytic and single-valued within and on C1 and does not have any
poles or zeros on C1, then the closed loop system (4.17) will be stable provided C2 does
not encircle the critical point (-1, jO).

The stability criteria of closed-loop AAS system as stated in the above theorem can be

assessed from the frequency response function, Gol(ja)) since the modified open-loop

45

transfer function, (301(5), is linear time-invariant, stable and proper. If the frequency

response function has a phase which is an integer multiple of -180°, for some frequency,

a) = (01, its steady state magnitude will be |G0,(jw)|, a real constant. The gain, K of the

AAS controller in that case can be increased by a factor of 1/lG01(jw)l before the curve

C2 encircles the point (—1,j0) and the closed loop AAS system becomes unstable. i is
a

the stability gain margin of the AAS system at the phase crossover frequency of col.

The phase crossings of the modified open-loop transfer function, G0,“) at -180°,

if any, are obtained by predicting the phase angles of individual transfer function

appearing in (301(3). The speaker used as an acoustic absorber when placed in an

enclosure is assumed to behave as a circular piston mounted in an infinite baffle. The

transfer function, Groom (s), between acoustic pressure component, pspkp and the speaker

volumetric flow rate, qspkr’ then follows from the following relation for a circular piston

(Beranek, 1986)

 

 

 

 

 

 

P . . c 110’" k ) ' c
Sp," (Jw)= Groom(1w)= L0 1‘ Sp r + mo 2 K1(2k’spkr) (4°20)
gspkr Aspkr krspkr 2(kAspkr)
where, k = 9—) is the wave number. J1 and K] are the Bessell functions represented by the
c
series
3 5 7
x x x x
J =__ + _ ...... (421)
l 2 22.4 22-42-6 22 42-62 -8
3 5 7
x x x
K (x =2 —— + ------ ) (4.22)
1 ) (3 325 3252.7

where x = Zkrspkr" rspk, is the effective radius of the speaker and equal to 66% of the

advertised nominal radius of the speaker. At low frequencies where krspk, << 2, the

reactive component of (4.20) is dominant and the first two terms of the J1(x), and K1( x)

series are sufficient to express it (Colloms, 1985). This simplifies the room transfer

46

function, Groom, to
8
Gm," (s) = (—,f—0—) s (4.23)
3” rspkr
The room acoustics thus has a constant 90° phase. The transfer function, Groom, was

measured with the enclosed, 12 inch, dual-wound coil subwoofer speaker (Radio Shack
model 40-1350) and compares reasonably well with the response of theoretical model
(4.23). The measured magnitude response has 20 dB/decade slope (Figure 4.3) and
measured phase response is close to 90° suggesting a transfer function of the form (4.23)
for room acoustics. For an enclosed speaker, the speaker cone motion is affected by the
enclosure at the back and the assumption that an enclosure acts as an infinite baffle is not
very accurate. This introduces inaccuracies in the theoretical model causing differences
between measured and theoretical response of room acoustics. The closed loop transfer

function, Tspk,(s), of the compensated speaker approaches a real constant at high

proportional gains and does not contribute to the phase of modified open-loop transfer
function. The sensitivity of the microphone, Smic’ and controller transfer function, GAAS
(4.16) are both real constants and hence have 0° phase. Therefore, the modified open-loop
transfer function theoretically has a constant 90° phase. It does not cross -180° at any
frequency giving an infinite gain margin for the closed-loop system. In theory, the active
acoustic sink controller model (4.16) will thus yield a stable closed-loop AAS system for
any 0 < K < co. It must be noted here that any unmodeled or unstable high frequency
dynamics of the compensated speaker and room acoustics will reduce the gain margin and

thereby limit the gain of AAS controller.

47

 

 

  

 

 

§—i— Theoretical
v—e— Measured '

0, 225532., :
100 220

 

 

Log-Magnitude (dB)
U!
o
l

 

 

 

 

 

 

 

 

 

20
Frequency (Hz)
200 I I I I I I E
g ‘ *_"‘"-‘."~'—~—-=-=-I 5 : 3 3 --_....1
°° E E E E E E
é o_ .......................................................... -
g , Theoretical
a. ----- Measured '
-200 i i ‘
20 100 220
Frequency (Hz)

Figure 4.3. Comparison of Theoretical and Measured Magnitude Response of Room

Acoustics (dB re 1 Pa-sec/m)

4.5. Summary

An analytical Active Acoustic Sink (AAS) controller model was developed in this
chapter to achieve an acoustically absorptive surface at the face of a speaker cone. It
provides acoustic power absorption over a wide range of noise frequencies by directing
local acoustic power flux into the speaker. A negative real acoustic impedance at the
speaker face is shown to be necessary for an acoustically absorptive surface. It was also
shown that speaker compensation is necessary for the successful performance of AAS. A

stability analysis was done to verify that the resulting AAS controller results in a stable

closed-loop system.
48

Chapter 5 Implementation and Testing of Active Acoustic Sink Controller

5.1. Introduction

The implementation and testing of the AAS system is discussed in this chapter.
The AAS controller gain is chosen using the gain margin of the open loop AAS system.
The AAS controller is implemented using a velocity compensated speaker to provide
maximum possible absorption of local acoustic power flux over 65-200 Hz bandwidth.

Experimental tests of the AAS indicate successful performance.

5.2. Implementation of Active Acoustic Sink Controller
The AAS controller is implemented with the controller gain obtained from the gain

margin of open loop AAS system. The open-loop configuration shown in Figure 5.1 is

obtained by breaking the measured pressure path and inserting a reference voltage, r( t ).

The frequency response of the modified open-loop transfer function,

60,“) = ( T,

pk,Aspk,G,00m, from reference signal, r(t), to microphone signal,

SmicAspkr J
KB

=1. An enclosed, compensated l2-inch diameter, dual-
mic spkr

ep, is measured for [

wound coil subwoofer speaker (Radio Shack Realistic model 40-1350) is used to measure

the frequency response of transfer function. The effective speaker area, A

spkr , 0f the

dual-wound subwoofer is 0.03 m2 and the value of 6 used was 6. The speaker face

pressure was measured by a microphone (Radio Shack Model 33-1052; sensitivity,

Smic = 4 mv/Pa ) placed at a distance of 18 cm from the speaker face. The measured

open-loop frequency response (Figure 5.2) shows an infinite phase margin and a 19 dB

49

 

gain margin at the phase—crossover frequency of 813 Hz. The AAS gain, K, can therefore
be increased from its open-loop value of 2.1x10'5 by a factor of 8.9 (19 dB) before
destabilizing the closed loop AAS system. A gain of 10'4 meets the recommended 6 dB
gain margin and 30-60° phase margin requirement for SISO closed loop system (Ogata,

1986) and was chosen to implement the AAS system.

 

 

 

Compensated Room
Speaker q Acoustics
e (1?) Va: ”spkr spkr E Ps kr
' Wig Tspkr room J g:
_. .g'

 

 

r( t)

 

 

 

 

3;" .i 1'“ , v
-.1—l'....r

 

 

 

 

 

 

  
    

 

. _ EPhase Cross-Over;
: : : : : :Fre uenc 813Hz
'2“) L I Z L I i q y 1( I )

45 100

 

 

 

Frequency (Hz) 1000 1645

Figure 5.2. Bode Diagram of Open Loop Active Acoustic Sink System (dB re 0.004)

50

The distance of microphone from the speaker face is critical in the selection of
AAS controller gain as it affects the gain margin of the system. As the microphone is
moved closer to the speaker face, the open loop gain of the system increases thereby
reducing the available gain margin. The microphone should therefore be moved away
from the speaker face to increase the AAS absorber gain. The AAS design objective
(4.12) requires the speaker volumetric flow rate due to speaker cone velocity track
acoustic pressure at the speaker face. The pressure measurement should therefore be
ideally made at the speaker face. Higher signal levels in the AAS loop also require that
microphone be placed closer to the speaker face. These conflicting requirements have to
be met while selecting the microphone distance. In our implementation AAS system,
microphone was placed 18 cm away from the speaker face to satisfy these requirements.
This distance between the microphone and the speaker surface gives rise to a measured
phase angle between pressure and speaker cone velocity that is different from the
physically correct one. For the propagating part of the sound field from the speaker face,
this phase difference can be estimated from the propagation speed of sound, c, and the

distance, Ax, between speaker cone and diaphragm of the microphone. The time required

by the sound wave to travel the distance Ax results in an estimated phase shift, on.

A¢x = (go-)xf x Ax (in degrees) (5.1)
c

The phase shift is proportional to frequency, f (Hz) and the distance, Ax. For a distance
of Ax =18 cm, used in our AAS system implementation, this phase shift is considerable
even at small frequencies as seen from Figure 5.3. A phase shift correction is therefore

necessary to evaluate AAS system performance at the speaker face.

5.3. Performance Evaluation of Active Acoustic Sink System
The performance of AAS system is evaluated by three independent methods:
speaker face impedance, the acoustic energy absorbed at the speaker face, and the P-I

index defined as the ratio of sound pressure level at the speaker face to acoustic intensity.

51

 

250 : ::'::::'

 

 

 

 

-50 i

 

Figure 5.3. Phase Shift, A¢x , due to Finite Distance between Microphone and Speaker

Cone

These three methods will be discussed in detail below.

The speaker face impedance, Z, is the transfer function from speaker volumetric
flow rate, qspkr’ to speaker face pressure, pface’ and provides the first criteria for
evaluating the effectiveness of the AAS system. It is in general a complex quantity at any
given frequency '

Z(f)=:fafi=R(f)+iX(f) (5-2)

spkr
The real part, R( f ), is the acoustic resistance and represents the in-phase, non-
conservative, energy transfer at the speaker face. The imaginary part, X( f), is the
acoustic reactance and represents the out-of-phase, conservative, energy transfer at the
speaker face. For negative acoustic resistance, acoustic energy will be absorbed at the
speaker face. The AAS objective (4. 12) requires speaker face impedance to be pure real

and negative for absorption of incident acoustic power.

-1
=_ 5.3
Z K ( )

Using (4.12) and (5.3), more acoustic energy will be absorbed at the speaker face as the
52

negative real speaker face impedance becomes smaller in magnitude. The speaker face
acoustic resistance should therefore be negative and speaker face acoustic reactance
ideally be zero for perfect absorption of acoustic power. Any non-zero, negative speaker
face resistance, will result in net energy absorption by the AAS independent of speaker
face reactance. I

Speaker face acoustic impedance was measured (Figure 5.4) for two independent
cases to evaluate the effectiveness of the AAS. An enclosed subwoofer used as noise
source generated acoustic noise of constant amplitude over the 65-200 Hz bandwidth of
the test. The measurement of disconnected speaker impedance with no applied control
(uncontrolled system) (Figure 5.5) has a 22 dB gain and 100° phase variation over 65—200
Hz bandwidth. The measurement of AAS impedance with the controlled, compensated
speaker (controlled system) (Figure 5.6) shows a 9.6 dB gain and 40° phase variation over
the same bandwidth. The acoustic impedance is nearly negative real and has smaller
magnitude over this frequency range for the controlled system indicating that it is possible
to absorb more acoustic energy with the controlled, compensated speaker. At frequencies
below 100 Hz, the real and imaginary parts of the speaker impedance are comparable for
both the controlled and uncontrolled systems. Therefore, a little change in the acoustic
energy absorbed from uncontrolled to controlled state is expected below 100 Hz. At
frequencies above 100 Hz, the imaginary part of the speaker impedance dominates for the
uncontrolled system while speaker impedance is nearly negative real for the controlled

system and a larger increase in energy absorption is expected.

53

Noise Source AAS System

    

Control 8 - aker

 

 

 

 

 

Log-Magnitude (dB)

65 100 Frequency (Hz) 200

 

 

Phase (Degrees)

 

J

0 : . .
65 100

 

 

Frequency (Hz) 200

Figure 5.5. Speaker Impedance of Uncontrolled AAS System (dB re Pa-sec/m3)

54

 

é

 

 

 

 

55.

i=3

En 50 _ ............................................................................. .
a

2.

DD

0 . .

A O J . . 1

65 100 Frequency (Hz) 200
400

 

I

 

Phase (Degrees)
N
8

 

 

 

65 100 Frequency (Hz) 200

Figure 5.6. Speaker Impedance of Controlled AAS System (dB re Pa-sec/m3)

The change in acoustic energy absorbed with frequency can be derived from the
relationship between variation in acoustic intensity with magnitude and phase of the
speaker impedance. The time-averaged acoustic intensity at the speaker face is the time-

averaged flow of acoustic energy across a unit surface area of the speaker. It can be

written as

T T
—1 1
I = t t d: = _ t t dt 54
2T_..'Tpface( )Vspkr( ) ZTAspkr —ITPface( )qspk,( ) ( )

where, pfaceh‘) is the instantaneous value of acoustic pressure, qspk,(t) is the

instantaneous value of speaker volumetric flow rate due to speaker cone velocity, vspk,( t )

measured in direction normal to speaker cone surface. For a sinusoidal variation of

55

magnitude a in the speaker face pressure, p face( I ) = 0130411150, the speaker volumetric
flow rate is, (Ispkrm = ysin(-2-]Tflt+ 41). y = UK is the amplitude of the speaker

volumetric flow rate. UK and (l) are the magnitude and phase of the speaker face

impedance at frequency of l/ T Hz. The acoustic intensity can then be written as

 

 

 

2 T
a K 27: 27:
I = ' —t ' —t+ dt 5.5
(f) ”Asp” Lard T )sm( T 4’) ( )
2 2
.°.I(f)= a chos¢= 7 xchos¢ (5.6)
2Aspkr spkr K

For a constant amplitude, y, of speaker volumetric flow rate, the change in acoustic
intensity with frequency is proportional to the change in speaker impedance magnitude
and phase. The variation in absorbed acoustic energy with frequency in controlled and
uncontrolled AAS system can thus be predicted from the speaker impedance curves
(Figures 5.5-5.6) and the phase correction curve (Figure 5.3). For example, the
uncontrolled impedance magnitude changes from 4731.5 (73.5 dB) to 8912.5 (79 dB)
from 111 Hz to 145 Hz. The phase angle is nearly constant at 268° after correction for
propagation phase shift at these frequencies. An increase of 2.75 dB in absorbed acoustic
energy is therefore expected from 111 Hz to 145 Hz. Over the same frequency range, the
controlled impedance magnitude changes from 2387.32 (67.56 dB) to 4774.65 (73.58 dB)
while the adjusted phase changes from 163° to 191°. An increase of 3.1 dB in absorbed
acoustic intensity is expected for controlled system from 111 Hz to 145 Hz.

Acoustic energy absorbed by the AAS system was measured to demonstrate its
performance (Figure 5.4). A Brfiel & Kjaer sound intensity probe (Type 3519) was used
to measure acoustic intensity at the front of the speaker for both the uncontrolled and
controlled systems. In each test, the acoustic power absorbed by the AAS was indicated
by the time-averaged acoustic intensity vector, 1, into the speaker. These two
measurements allowed an independent evaluation of the control performance because the

56

 

acoustic energy absorbed in the uncontrolled state was only due to the mechanical
dissipation of the speaker.

Acoustic intensity measurements below 70 Hz were demonstrated unreliable due
to small pressure signal levels and are not included here. At these frequencies, the
measured open-loop response magnitude is less than -45 dB (Figure 5.2). This makes the
effect of noise dominant below 70 Hz and accordingly measurements below 70 Hz were
found unreliable due to poor signal-to-noise ratio and verified by poor measured

coherence below 70 Hz (Figure 5.7).

 

Coherence
:3
UI
r
i

 

 

 

Frequency (Hz)

Figure 5.7. Measured Coherence in the Open Loop Frequency Response of AAS

The acoustic energy absorbed in the controlled state by the AAS (Figure 11 and
Table 5.1) shows a substantial increase over the uncontrolled state. The percent increase
ranges from 12.95% over 70.5-80 Hz to 143.78% over 170.5-180 Hz. This was expected
since the impedance of controlled speaker is more nearly negative real over a wider
frequency range (Fig. 9). The measured acoustic energy absorbed from 111 Hz to 145 Hz
increases by 2.74 dB in uncontrolled state and agrees well with the 2.75 dB increase
predicted from the uncontrolled speaker impedance measurements. Over the same

frequency range, the increase in acoustic energy absorbed in controlled state is 2.4 dB and

57

is close to an increase of 3.1 dB predicted from the controlled speaker impedance.
The measurement of P-I index is the final method to check the effectiveness of the

AAS system. It is defined as

"2'
6p, = IOlogI:BI—] - lOlog[pc]' (5.7)

It is related to the ratio of sound pressure level to intensity and allows the evaluation of the
AAS system with reference to the speaker face pressure. The increase in absorbed
acoustic energy from uncontrolled to controlled state was calculated from the P-I index
(Table 5.2). It compares well with the increase in absorbed acoustic energy measured by
the Briiel & Kjaer sound intensity probe.

The acoustic intensity measurements demonstrate the effectiveness of AAS and
verify the requirement that speaker impedance must be negative real for acoustic energy to
be absorbed into the speaker. The acoustical effect of this would be a high absorption of
acoustic energy into the speaker. The AAS system developed in this work can provide an
alternate solution to the problem of noise reduction in an acoustic space. Several acoustic

sinks can be placed in an acoustic space to absorb acoustic energy.

58

Table 5.1 Acoustic Energy absorbed by the Active Acoustic Sink

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

   

Frequency Acoustic Acoustic Net Increase Percent
Band Intensity for Intensity for in Acoustic Increase
Uncontrolled Controlled Intensity
(Hz) system (p W/mz) system ( pW/mz) ( pW/m2 )
70.5-80 3981.07 4496.763 515.69 12.95 %
80.5-90 3311.31 3993.93 682.61 20.61 %
90.5-100 2460.37 2726.47 266.09 10.82 %
100.5-110 2902.69 4971.64 2068.96 71.28 %
110.5-120 5318.63 12589.25 7270.62 136.70 %
120.5-130 7216.06 19760.59 12544.54 173.84 %
130.5-140 8757.90 25038.02 16280.12 185.39 %
140.5-150 10000.00 21877.62 11877.62 118.78 %
1505160 6444.66 17422.08 10977.42 170.33 %
160.5-170 4075.68 11220.185 7144.51 175.30 %
170.5-180 3512.37 8562.4885 5050.12 143.78 %
5:: 44.00
: 42.00
'5 40.00
g 38.00
:5 36.00 El Uncontrolled
.3 34.00
g 32.00 I Controlled
2 30.00 ; . .
o - I.
o,‘ 8 2 O o O ‘ .
E 3 v5 73 9.1 3.” 1T E E E3 g
00 o ' V7 W V} I 1 T -—1
ex 8 g o' o' 2 V? V) .5
”~92:§§9
Frequency (Hz) —‘

Figure 5.8. Acoustic Intensity Absorbed by AAS in uncontrolled and controlled states

59

Table 5.2 Increase in Absorbed Acoustic Intensity from Uncontrolled to Controlled States

 

Frequency From P-I Index From Intensity
Band (dB) Probe (dB)

 

 

 

 

 

 

 

 

 

 

 

1005110 2.34 2.04
1 105-120 3.74 4.06
1205-130 4.37 i 3.61
1305140 4.56 5.15
140.5150 3.4 4.15
150.5-160 4.32 4.86
160.5-170 4.4 3.23
170.5-180 3.87 4.24

 

 

 

 

 

5.4. Summary

The AAS system was constructed and its performance evaluated using a velocity
compensated, enclosed, dual-wound subwoofer speaker in this chapter. The effectiveness
of AAS system was evaluated by measuring speaker impedance in uncontrolled
(disconnected speaker with no applied control) and controlled states (controlled,
compensated speaker). The measured impedance phase was corrected to account for
phase shift introduced due to microphone-laser velocimeter configuration used to measure
impedance. The controlled speaker impedance magnitude varied by 9.6 dB and its phase
changed from 165° to 200° after correcting for propagation phase shift over 65-200 Hz

60

 

bandwidth. Over the same bandwidth, the magnitude of uncontrolled speaker impedance
varied by 22 dB gain and its adjusted phase varied from 188° to 267°. The controlled
speaker impedance is thus nearly negative real indicating it is possible to absorb more
acoustic energy with controlled, compensated speaker. The effectiveness of AAS is
demonstrated by measuring the increase in acoustic energy absorbed by the uncontrolled
and controlled compensated speaker. The results show an increase from 12.95 % over
70.5-80 Hz to 175.3% over 160.5-170 Hz indicating that it is possible to use a

compensated speaker for sound power absorption from an acoustic volume.

61

Chapter 6 Conclusions

6.1. Dissertation Summary

A prototype Active Acoustic Sink (AAS) was designed and constructed to
determine if an electronically driven speaker can be built to absorb acoustic energy. To
guarantee the flow of acoustic energy into the speaker, the speaker cone velocity must
accurately track acoustic pressure in front of the speaker. The varying velocity frequency
response of speakers makes them unsuitable as energy absorbers unless they are
compensated. Appropriate compensator design requires an accurate speaker dynamic
model.

A simple laboratory based methodology for developing a suitable speaker dynamic
model has been presented and experimentally verified in laboratory tests on a dual-coil
subwoofer speaker. The speaker cone velocity frequency response is shown to have large
magnitude and phase variations below 200 Hz. This makes the subwoofer speaker
unsuitable actuator for AAS without compensation. This is the first time that a dynamic
model of dual-wound coil subwoofer has been developed to include the effect of mutual
inductance between dual coils.

Speaker velocity feedback compensation for minimizing the magnitude and phase
variations in velocity response has been designed and experimentally verified in
laboratory tests on a dual-coil subwoofer speaker. It uses a proportional controller to drive
the subwoofer through primary speaker coils. The speaker cone velocity sensing is done
by a novel velocity sensor designed using a bond graph model of the subwoofer. It uses

speaker cone motion induced secondary coil voltage obtained by compensating the mutual

62

inductance effect between dual-wound coils. The compensated speaker has variations less
than 5 dB in gain and 20 degrees in phase over 4400 Hz bandwidth compared to 20 dB
and 160 degrees variations for the uncompensated speaker. This is the first successful
implementation of speaker velocity feedback compensation to date which does not require
any external sensor for measuring speaker cone velocity. Compensation of speaker
response now allows the speaker volumetric flow rate to accurately follow the speaker
face pressure.

An analytical AAS controller model was developed using the compensated speaker
model. The controller design objective was to yield an acoustically absorptive speaker
cone surface by generating negative real acoustic impedance at the cone surface of
compensated speaker. The resulting controller is a pure gain and a closed loop stability
analysis was done to determine its maximum stabilizing value. A stability analysis of
point feedback acoustic absorption system has been done for the first time in this work.

The AAS system was constructed and its performance evaluated using a velocity
compensated, enclosed, dual-wound subwoofer speaker. The effectiveness of AAS
system was evaluated by measuring speaker impedance in uncontrolled (disconnected
speaker with no applied control) and controlled states (controlled, compensated speaker).
The measured impedance phase was corrected to account for phase shift introduced due to
microphone-laser velocimeter configuration used to measure impedance. The controlled
speaker impedance magnitude varied by 9.6 dB and its adjusted phase changed from 165°
to 200° over 65-200 Hz bandwidth. Over the same bandwidth, the magnitude of
uncontrolled speaker impedance varied by 22 dB gain and its adjusted phase varied from
188° to 267°. The controlled speaker impedance is thus nearly negative real indicating it
is possible to absorb more acoustic energy with a controlled, compensated speaker. The
effectiveness of AAS is demonstrated by measuring the increase in acoustic energy
absorbed by the uncontrolled and controlled compensated speaker. The results show an
increase from 12.95 % over 70.5-80 Hz to 175.3% over 160.5-170 Hz indicating that it is

63

 

possible to use a compensated speaker for sound power absorption from an acoustic
volume. This is an improvement over the results obtained by Hall (1994) with his AAS
which was limited due to ineffective speaker cone velocity sensor. The AAS uses simple
control technology, is easy to implement, and does not require any changes in the physical

parameters of the system components.

6.2. Directions for Future Work

There are several areas for future work or improvement in the method stated in this
work. The most obvious next step would be to test the effectiveness of AAS in a three-
dimensional enclosure such as a vehicle interior. Effective three-dimensional noise
control can also be beneficial in the aircraft industry and factories.

The performance of AAS can be improved further by investigating different AAS
control algorithms. The weak signal levels at low frequencies in the AAS loop did not
allow for good coherent measurements below 65 Hz. The AAS controller can be modified
to increase the open loop low frequency gain of the AAS system without changing the
high frequency behavior. This might improve the low frequency performance of the
system. Another area of improvement is in the measurement of local acoustic intensity at
the speaker face of AAS system at low frequencies. The two microphone probe used to
measure the intensity cannot be used effectively below 40 Hz. The amount of acoustic

power absorption can also be increased by placing more AAS systems.

 

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68

 

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