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Michigan State
University

 

 

 

 

 

 

 

 

 

 

 

 

This is to certify that the

thesis entitled

A Compensated Actuator for
an Acoustic Duct

presented by

Charles Birdsong

has been accepted towards fulfillment
of the requirements for

Masters degree in M1311], Engineering

 

Major professo

Date 12/10/96

 

@7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

PLACE N RETURN BOX to remove this checkout from your record.
TO AVOID FINES rotum on or before date duo.

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MSU loAn Affirmative Halon/EN Opportunly Imam

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A COMPENSATED ACTUATOR
FOR AN ACOUSTIC DUCT

By

Charles Birdsong

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

MASTER OF SCIENCE

Department of Mechanical Engineering

1996

ABSTRACT

A COMPENSATED ACTUATOR
FOR AN ACOUSTIC DUCT

By

Charles Birdsong

Audio speakers are commonly used as acoustic actuators for noise control
applications. Recent developments in the use of compensated dual-coil
speakers have improved the performance of these acoustic actuators.
However, the performance of these speakers depends on the application.
When they are applied in systems with strong coupling between the plant
and the actuator such as an acoustic duct, the velocity sensor used in previous

work must be improved.

This study considers the application of a compensated speaker as an
actuator in noise control of an acoustic duct. The speaker dynamics and the
acoustic duct dynamics are first modeled separately. The two systems are then
coupled, and the resulting system is modeled. A velocity sensor is developed
and used in feed-back compensation. The resulting speaker system behaves
as an ideal actuator with minimal magnitude and phase variation over a O —
200 Hz bandwidth. These conclusions are verified through experimental

results.

This study is important in the overall area of acoustic actuators and active
noise control. The actuator developed here will significantly aid in the goal

of active noise control in an acoustic duct.

Capyrisht by
Charles Black Birdsong
1996

DEDICATION
I dedicate this work to my loving wife Bernadette whose support and

understanding have given strength.

iv

ACKNOWLEDGMENTS

Special thanks to those who came before me and paved the way for this work:
Dr. Clark Radcliffe, Sachin Gogate, and Andrew Hull. I would also like to
thank Brooks Byam, Giuseppe Derose, and Mark Minor for their day-to—day

support and assistance.

TABLE OF CONTENTS

Page

1. INTRODUCTION ....................................................................................................... 1
2. ACOUSTIC ACTUATOR SPEAKER MODEL ...................................................... 5
Identification and Modeling of Speaker Dynamics ..................................... 6
Speaker Model Transfer Functions ................................................................. 9
Velocity Feedback Compensation of Speaker ............................................. 11
Speaker Model Verification ............................................................................ 14
Speaker Dynamics ................................................................................. 15

Velocity Sensor ...................................................................................... 16

Velocity Feedback Compensation of Speaker ................................. 17

3. ACOUSTIC DUCT SYSTEM MODEL ................................................................... 20
System Model ..................................................................................................... 21

State Space Representation ............................................................................. 22

Duct Transfer Function .................................................................................... 23

Duct Model Verification .................................................................................. 24

4. COUPLED SPEAKER-DUCT SYSTEM ................................................................. 28
Modeling of Coupled Speaker-Duct System ................................................ 28
Velocity Sensor ...................................................................................... 31

Coupled Speaker-Duct Model Verification ................................................. 32

vi

Speaker Velocity Model ....................................................................... 32

Velocity Sensor ...................................................................................... 34
Velocity Feedback Compensation of Speaker ................................. 35
5. CONCLUSIONS ........................................................................................................ 37
APPENDIX A. DERIVATION OF SPEAKER MODEL STATE
EQUATIONS ..................................................................................... 38
APPENDIX B. SPEAKER PARAMETER IDENTIFICATION .............................. 43
APPENDIX C. DERIVATION OF SPEAKER MODEL TRANSFER
FUNCTIONS ..................................................................................... 47
APPENDIX D. COMPENSATOR CIRCUIT IMPLEMENTATION ..................... 50
APPENDIX E. ACOUSTIC DUCT STATE EQUATION
COMPUTATION ALGORITHM ................................................... 53
APPENDIX F. COUPLED SPEAKER/ DUCT VELOCITY SENSOR
TRANSFER FUNCTION ................................................................ 54
LIST OF REFERENCES ................................................................................................. 56

vii

LIST OF TABLES

Page
Table 3.1. Speaker Compliance Data ........................................................................ 43
Table 8.2. Speaker System Physical Parameters ..................................................... 46

viii

Figure 1.1.
Figure 1.2.

Figure 1.3.
Figure 2.1.
Figure 2.2.
Figure 2.3.
Figure 2.4.
Figure 2.5.

Figure 2.6.

Figure 2.7.

Figure 3.1.

Figure 3.2.
Figure 3.3.

Figure 4.1.

LIST OF FIGURES

Page

Typical Noise Control System ................................................................. 1
Speaker Face Velocity to Primary Coil Voltage Using

Original Compensator ............................................................................... 3
Acoustic Duct and Compensated Actuator Setup ............................... 4
Dual Voice-Coil Speaker Diagram .......................................................... 6
Bond Graph of Dual Voice-Coil Speaker System ................................ 7
Proportional Feedback Controller ........................................................ 12
Dual Voice-Coil Speaker and Velocity Transducer .......................... 14
Comparison of Theoretical and Experimental

Speaker Velocity Response .................................................................... 15
Comparison of Velocity Sensor and Measured

Velocity ....................................................................................................... 17
Measured Compensated Closed-Loop Velocity

Response of Speaker for Proportional Gain of 1.0, 10,

and 50 .......................................................................................................... 18
Acoustic Duct Experimental Setup with Laser

Velocity Transducer and Microphone ................................................. 25
Experimental Acoustic Duct System .................................................... 25
Comparison of Model and Measured Velocity to Duct

Pressure Transfer Function for Acoustic Duct ................................... 26

Coupled Speaker-Duct System ............................................................... 3O

ix

Figure 4.2. Coupled Speaker/ Duct Experimental Setup with
Laser Velocity Transducer and Velocity Sensor ................................ 32

Figure 4.3. Comparison of Measured Frequency Response for
Coupled Duct-Speaker System with Model ........................................ 33

Figure 4.4. Diagram of Velocity Sensor in Speaker/ Duct System
Implementation ....................................................................................... 34

Figure 4.5. Comparison of Measured, Modeled and Sensor
Model Speaker Velocity for Coupled Speaker/ Duct
System ......................................................................................................... 35

Figure 4.6. Comparison of Closed-Loop Speaker Velocity to
Primary Speaker Voltage Frequency Response for

Open Loop and Proportional Gains of 10, 20, and 100 ..................... 36
Figure A.1. Bond Graph of Dual Voice-Coil Speaker System ............................. 38
Figure B.2. Frequency Response of Speaker Face Velocity to

Primary Coil Voltage ............................................................................... 44
Figure D.1. Proportional Feedback Controller ....................................................... 51
Figure D.2. Speaker Compensation Circuit Diagram ........................................... 51

Figure D.3. Overall Arrangement of Compensated Speaker
Components .............................................................................................. 52

1. INTRODUCTION

Active noise control is an expanding field in the automotive and aircraft
industries. Commercial products are currently available to create quiet
interior spaces [Bradley 1995, Warner 1995]. These systems use passive and
active controls to treat unwanted noise. Passive control consists of applying
dampening material to treat high frequency noise. Dampening material must
be of the same physical dimension as the wavelength of the sound wave to be
effective [General Motors 1994]. Below 200 Hz the wave length in air is
approximately 0.26 m or longer. This would require an unrealistic amount of
dampening material approximately one quarter of a meter thick for effective
passive noise control. For low frequency noise, active controls methods can
be used. Active control relies on a combination of sensors, a controller, and
actuators to treat noise in the system (plant). Figure 1.1 shows a typical active
noise control system where a microphone is used as a sensor and an audio

speaker is used as an acoustic actuator to treat noise in an interior space.

 

Actuator Plant
(Speaker) (Interior Space)

 

 

 

 

Sensor
(Microphone) I]

—| Controller 1%

Figure 1.1. Typical Noise Control System

 

 

One system that has received much attention in this field is the acoustic
duct, which consists of a long, hard walled enclosure. Hull showed that the
resonances excited by a noise source in an acoustic duct can be attenuated
using feedback active noise control [Hull 1993]. Attempts at wide band noise
control were hindered by actuator dynamics that caused the measured

control input to deviate from the desired control.

Gogate proposed a strategy for eliminating the effects of speaker dynamics
through feed-back compensation [Radcliffe et a1. 1996]. The original design
did not include the effects of the coupled dynamics through the interaction of
the plant pressure and the actuator. Figure 1.2 shows the speaker face velocity
to primary coil voltage frequency response of the original compensator with
two cases: the dashed line represents the response of the speaker in free-air,
and the solid line represents the response when the speaker is coupled with
an acoustic duct. In free-air there is little magnitude and phase variation
from O - 200 Hz. When the speaker is coupled with the acoustic duct, there
are large magnitude and phase variations exhibited at the resonance
frequencies of the duct. The original compensator fails to eliminate the

dynamics associated with the duct.

 

 

 

 

 

i i 1 .
50 100 150 200 250 300 350 400
Frequency (Hz)

 

 

 

 

 

i . . i
50 100 150 200 250 300 350 400
Frequency (Hz)

Figure 1.2. Speaker Face Velocity to Primary Coil Voltage
Using Original Compensator

This study presents an acoustic actuator that compensates for both
actuator and plant dynamics. The acoustic duct is presented as a plant in
order to demonstrate the robustness of the system to a plant with strong
coupling with the actuator through pressure interaction. A model of a dual
voice-coil speaker is first presented, and a velocity sensor is developed. It is
shown that the speaker dynamics can be eliminated through feed-back
compensation. A model of an acoustic duct is presented next, which predicts
the pressure response due to a velocity input. Finally, the two systems are
coupled, and it is shown that the speaker compensation minimizes both the
speaker and the acoustic duct dynamics through feed-back compensation.
The results in every stage of the modeling and analysis are verified through
experimental testing, and model results are presented together with
experimental results. Figure 1.3 shows the experimental setup used to verify

the model results.

 

Figure 1.3. Acoustic Duct and Compensated Actuator Setup

The work presented here provides a method for creating an ideal acoustic
actuator for systems that include strong plant and actuator coupling and
brings the goal of active noise control of systems such as the acoustic duct one

step closer.

2. ACOUSTIC ACTUATOR SPEAKER MODEL

Audio speakers are commonly used as acoustic actuators in noise control
systems. They are beneficial because a small voltage applied to a speaker can
generate a strong control effort. Audio speakers are relatively inexpensive
and widely available in commercial sizes and models. Speakers have the
disadvantage that the response of a speaker can be strongly affected by both
the dynamics associated with the free-air resonance of the speaker and the
dynamics of the system it is driving. An ideal actuator will have a pure gain
over the required bandwidth. When a speaker is affected by dynamics, it can
exhibit significant magnitude and phase variations limiting its performance.
If a speaker is to be used as an acoustic actuator, these effects must be

minimized.

One method of minimizing magnitude and phase variations is to apply
feedback compensation to the speaker. If the speaker response can be
measured, then the signal can be applied to a feedback controller, the response
can be driven to the desired output, and the magnitude and phase variation

can be reduced. An accurate speaker face velocity sensor is therefore required.

The development of a speaker, velocity sensor model and the system
transfer functions follows. The model is then used to develop a feedback
compensation strategy. Finally, the model is verified through experimental

testing.

Identification and Modeling of Speaker Dynamics

One variety of speaker named the ”dual voice-coil” speaker has certain
characteristics that make it ideal for use as an acoustic actuator [Radcliffe et a1.
1996]. The dual voice-coil speaker has 2 independent wire coils intertwined
and wrapped around a bobbin which is allowed to slide over a permanent

magnet. This configuration is shown in Figure 2.1.

Primary Coil Bobbin
Terminal

  

 

Permanent .
Magnet \ ff Air
Cone
Secondary Coil
Teminal

Figure 2.1. Dual Voice-Coil Speaker Diagram

The electromagnetic speaker is a system that involves mechanical,
electrical, and magnetic energy domains. This class of problem is well suited
to modeling using the Bond Graph technique [Karnopp et a1 1990]. Figure 2.2
shows the bond graph diagram of the dual voice-coil speaker system [Radcliffe
et a1. 1996]. The previously available bond graph model is used here with the
addition of one ’R’ element which represents a current sensing resistor. New
velocity transfer functions are derived from the bond graph model and used

in the speaker feedback control system.

coil

Secondary Coil

bl
Sflcbs >1Iebsi > GY 15p,“

is=cbs;b=o F
SD

Icoil 1 Ll—ATF |_;

Q0)
Primarys Coil bl

6" PEA GY Cspkr

Figure 2.2. Bond Graph of Dual Voice-Coil Speaker System

The Bond Graph arrows represent power flow from the electrical voltage,

5,, applied at the primary coil through the model’s elements. Power is
dissipated in resistive elements Rm“, Ru1 and Rm, 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 ’C’ elements. The open circuit on the
secondary coil is represented by a secondary coil current (i, =0) imposed by the

source of flow, 8,.

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

2p and 2.5; speaker cone displacement, xsph; and speaker cone velocity, A

v‘spkr
linear approximation is used to relate the flux linkages, AP and 113, of the
multiportl a,” field to the port currents, i and i,, in the primary and secondary

coils.

The state equations for the speaker model can be written from the bond
graph in state space form by choosing energy variables as state variables and

appropriate input and output variables. A complete derivation is given in

 

 

 

 

 

 

 

Appendix A.
5r(t) = Aspk,x(t) + Bspkruspkru) (2.1)
where
' 0 1 0 _ [0 0 '
—1 -R.rpkr bl 0 -SD
Aspkr = CSPk’I-‘Pk’ Ispkr (IspkrIcoil) I Bspkr = Isph (2.2)
— . 1 0
0 _bl (Rem! + Rm)
_ Icoil _ u _
x57,”
The state variables are xsp,” = Vs» , speaker face displacement, xspkr; speaker
A

P

face velocity, vspkr; and primary coil flux, AP. The input variables are

e
uspkr = [1:]; speaker primary coil voltage, ep; and speaker face pressure, P.

The output equations for the speaker model can be arranged in state

equation format as

yspkr(t) = Csperspkr(t) + DspkruSpkr(t) (23)

ebs

where the output variables, yspk, = in are speaker secondary coil voltage,

vspkr

e,,(t); speaker drive current in the primary coil, i,,(t); and speaker face velocity,

v The output matrices are

spkr'

 

 

 

' . M . l
0 bl 1-—Cfll- "Fifi..." +Rm) "M l
I coil coil I C O
1 c
Crpkr = O O 7— ’ Drpkr = O O (24)
cor! 0 O
_o 1 o _ ' ‘

 

 

Equations (2.1) and (2.3) define the model of a dual voice-coil speaker.
The speaker parameters necessary to define the model are the mechanical

inertia of speaker, 1W; mechanical compliance of speaker, CW; viscous
friction of speaker, RM; electromagnetic coupling factor, bl; speaker coil

resistance, R

CO

,,; speaker coil inductance, Icou, mutual inductance, M H; and the

C0

equivalent speaker area, SD. With the exception of mutual inductance, Mm”,

these electrical and mechanical parameters are defined in IEEE standard 219-

1975 [IEEE Standard 219-1975] for loudspeaker measurements.

Speaker Model Transfer Functions

The transfer function matrix can be determined from (2.1) and (2.3). The
speaker model includes two inputs and three outputs, resulting in a transfer

function matrix of dimension 3x2. The steady-state output of the system is

given by [Rough, 1996]

10

Y(s) = [C(sI - A)”1 B + D]U(s) (2.5)

where A is the state matrix; B is the input matrix; C is the output matrix; D is
the direct feed-through matrix; 'sI’ is the Laplace variable multiplied by an
identity matrix, U(s); and Y(s) is are the input and output vectors in the

Laplace domain.

The transfer function matrix is defined by

G(s) = C(sl — A)" B + D (2.6)

Substituting the values of A, B and C, inverting sI-A and multiplying the

matrices yields

e.b3 (S) Gebs/ ep Gebs/ P ep (S)
5(3) = Gm 0W P(s) (2.7)
vspkr (S) Gvspkrl ep Gvspkrl P

where each element in the transfer function matrix G(s) is given by

C I M. 3+C. R M. 2+b12C. +M..
Gem/6p =[ splcr spkr corl)s ( .spkr spkr coal)s ( spkr “"0514!“ (28)

2 R
G = [(CspkrIspkr )S + (Csplrr spkr )S + %en (29)

ip/ep

_ [(191 C

S
Gvspkr/ep _ Wk')%kn (2.10)

11

_ _ 2
abs/P = [ bl CIPkrSD[(ICoiI Alcoil)S + Rcoils%m (211)
0.,” = [W SDC‘P*’)%M (2.12)
2
Gvspkr/P = [_SDC:pkr[IcoilS + RcoilSH/de (213)

and the denominator of the G(s) matrix is given as

G,“ = (C,l[,,,,13p,,,lm,.,)s3 +(CWIWR + C I Rw,‘,)s2 +

coil .rpkr coil .
(blzCspk, + 1,0,, + C R Rm,» + R,

spkr coil oil

(2.14)

Equations (2.10) and (2.13) are new and important results not used in
previous work. All of the speaker model transfer function equations will be
useful when designing a velocity sensor for the speaker and when modeling

the speaker coupled with the acoustic duct.

Velocity Feedback Compensation of Speaker

The velocity of the speaker, Vspm is strongly affected by the dynamics of
the speaker and the pressure input, P. These effects will combine to create
magnitude and phase variations in the primary coil voltage to speaker
velocity response. One method of eliminating these unwanted effects is to
apply a proportional feedback controller as shown in Figure 2.3. The transfer
function for this system is given by (2.15), where Kp is the proportional gain

and H(s) is a velocity sensor. If the sensor transfer function is a real constant,

12

k, over the controller bandwidth, then the closed loop transfer function, T(s),
will approach a constant, 1/ k with zero phase. This compensation forces the
speaker cone velocity to accurately follow the desired velocity input. The
result is independent of the speaker dynamics and the input pressure

provided that the sensor has a constant transfer function over the controller

 

 

  

 

 

 

 

 

 

 

 

 

 

 

bandwidth.
vd(t) Proportional 6p“) Speaker vwk,(t)
Controller —> Dynarmcs ,
KP Gspkr
Velocity
Sensor
11(3)

 

 

 

Figure 2.3. Proportional Feedback Controller

 

T (spilt—£09.: KPG‘P"(S) (2.15)
’P‘" V,(s) 1+KPGW(s)H(s)

As Kp is increased, the transfer approaches 1/H(s) and the magnitude and
phase variation approaches zero. This approach assumes that the velocity of
the speaker face can be measured. A speaker velocity sensor is therefore
needed which accurately predicts the speaker velocity in the presence of

speaker and plant dynamics.

The relation between the speaker velocity and the two other measurable

outputs (the secondary coil voltage, e... and the primary coil current, i,) is

13

given in (2.7). The speaker velocity, V

spk, can be solved for in terms of 15,, and

I, yielding,
Vspk,(s) = HbsEb5(s) - Hp(s)Ip(s) (2.16)

where Hbs = 1 / bl and I-Ip(s)= sM / bl.

coil

The secondary coil voltage, E“, can be measured directly from the speaker
coil; the primary coil current, 1,, can be determined from the voltage across
the resistor, R"; and Hbs is a pure gain (1 / bl). The term Hp(s) is a differentiator
transfer function because it contains an ’s’ in the numerator. Such a transfer

function cannot be realized exactly, but an approximation Hp“) can be used

where

i1(s)=Mw"[ S ] (2.17)
” bl s-i-pl

 

where pl is a pole location selected such that Hp(s) approximates Hp(s) over

the controller bandwidth.

Feedback compensation can now be implemented using the signal from
the velocity sensor to compute the error between the desired velocity and the
sensor velocity and a proportional controller to drive the speaker velocity to
the desired velocity. It should be noted that the development of the velocity
sensor did not assume that the pressure at the speaker face was constant, as in

previous work. This new velocity sensor includes the effects of pressure as an

14

input to the system. As a result, the closed-loop system minimizes
magnitude and phase variations from not only the speaker dynamics (as in
previous work) but in addition, the dynamics associated with the acoustic
system, coupled through the pressure interaction with the speaker are
minimized as well. This improvement over the previous velocity sensor is
essential for the speaker to perform as an ideal actuator in a coupled system

such as the acoustic duct.

Speaker Model Verification

The system model was verified through experimental testing. The open-
loop speaker system was assembled, and the system response was compared to
the model. A velocity sensor was then created and tested; and finally, the
closed-loop system response was compared to the model. Figure 2.4 shows

the dual voice—coil speaker in an enclosure with the velocity transducer.

 

Figure 2.4. Dual Voice-Coil Speaker and Velocity Transducer

15

Speaker Dynamics
A 6-inch dual voice-coil speaker was mounted in an enclosure and the

physical parameters of the speaker system were determined [Radcliffe 1992].

The speaker velocity to voltage transfer function was computed (2.7) with
the speaker face pressure, P set equal to zero. This assumption was based on
the fact that when the speaker operates in free air, the pressure variation on
the speaker face is negligible. The speaker velocity to voltage transfer function
was then measured experimentally with a Hewlett Packard Signal Analyzer
35660A. The speaker face velocity was measured with a Bruel & Kjaer Laser
Doppler Velocity-Transducer Set Type 3544. Figure 2.5 shows the close
correlation with the experimental and model results. The solid line
represents the model, and the dotted line indicates the experimental

response.

 

Gain dB

 

 

 

 

200

 

Phas deg

 

 

 

 

-200 2 2 s 222221 a 2 gig-.13
101 10" 10
Frequencvfl-lz)

— — , Measured Model

 

Figure 2.5. Comparison of Theoretical and Experimental
Speaker Velocity Response

16

The speaker velocity response is affected by the dynamics of the free-air
resonance. There are approximately 20 dB and 180 degrees in magnitude and
phase variation from 1 - 200 Hz. While (2.7) shows that the transfer function
is third order, Figure 2.5 suggests that it can be modeled by a second order
system below approximately 200 Hz. The effective natural frequency of the

system can be calculated as

1
C I

.rpkr :pkr

 

f...) = = 6sz (2.18)

which matches the resonance frequency in Figure 2.5. At frequencies above
200 Hz, the response deviates from the 2nd order response due to inductance
effects. An ideal actuator will behave as a pure gain over a given band-width.
These results show that the speaker is an ineffective actuator when operated

in open-loop.

Velocity Sensor

An analog compensation circuit was built to implement the velocity
sensor given by (2.16). The differentiator was implemented using an op-amp
differentiating circuit with a high frequency pole at p1 = 1000 Hz. The sensor
velocity, vm to primary coil voltage, e, was then measured experimentally
and compared to the actual velocity measured by the velocity transducer.
Figure 2.6. shows the measured velocity compared with the modeled and

measured sensor response.

17

 

 

Magnitude dB

 

 

 

50 100 150 200 250 300 350
Frequency (Hz)

 

! I I 1 ' I I

 

Phase (Deg)

 

 

 

 

50 100 150 200 250 300 350
Frequency (Hz)

Figure 2.6. Comparison of Velocity Sensor and Measured Velocity

The velocity sensor gives good results below approximately 200 Hz.
There is with less than 2 dB magnitude and 20 degrees phase difference in this
frequency band. Minor magnitude and phase variation is exhibited below 20
Hz, and a significant phase variation is exhibited above 200 Hz. These errors
above 200 Hz are due to inductance effects in the speaker which become
significant at higher frequencies causing the degradation of the velocity
sensor response. These results indicate that the effective bandwidth of the

velocity sensor is 0 - 200 Hz.

Velocity Feedback Compensation of Speaker

A proportional controller was applied to the system using the velocity

sensor voltage as the feedback signal. The velocity response was measured

18

with different values of proportional gain. Figure 2.7 shows the closed-loop
velocity response measured with the velocity transducer with a gain of 1.0, 10

and 50.

 

  

 

 

.5
o
I

Magnitude dB
I “C, hI3 I

 

 

 

200 250 300

50 100

 

. Phase (Deg)

 

 

 

 

i i i i
50 100 1 50 200 250 300
Frequency (Hz)

Figure 2.7. Measured Compensated Closed-Loop Velocity Response of
Speaker for Proportional Gain of 1.0, 10, and 50

As predicted, higher gains reduced the magnitude and phase variation.
With a gain of 50, there is less than 5 dB magnitude and 35 degrees of phase
variation from 0 to 200 Hz, compared with 20 dB and 180 degrees magnitude
and phase variation for the uncompensated system. As expected, the
response above 200 Hz is degraded due to the error in the velocity sensor

response.

The results of the compensated speaker using the improved speaker
velocity sensor are comparable and exceed the results obtained using the

previous version of the velocity sensor [Radcliffe et a1. 1996]. The largest

19

magnitude variation shown in Figure 2.7 above 20 Hz is approximately 4 dB,
and the largest phase variation is approximately is 45 degrees at 200 Hz. This
is an improvement over the previous version of the compensator where the
largest variation was 5 dB magnitude and 90 degrees phase with a

proportional gain of 50.

The close agreement between the experimental and model results
indicate that the model accurately predicts the response of the system.
Furthermore, the velocity sensor results demonstrate that the actual velocity
of the speaker can be measured through the circuitry of the velocity sensor.
Finally, minimization of the magnitude and phase variation in the closed
loop response of the speaker indicates that the dynamics of the free-air
resonance of the speaker can be reduced through feedback compensation.
Together, these results show that the compensated speaker is an effective
acoustic actuator with essentially a pure gain from 0 to 200 Hz. This actuator

can now be applied to an acoustic system.

3. ACOUSTIC DUCT SYSTEM MODEL

The velocity sensor developed in the previous section has an important
feature that was not present in the original velocity sensor. It can predict the
speaker response when the velocity is caused by both a primary voltage input
and a pressure input. The original velocity sensor [Radcliffe et al. 1996] was
based on the assumption that the pressure input was zero, resulting in a
sensor that could only predict the velocity if it was caused by a primary
voltage input and the pressure on the speaker face was zero. This
improvement results in an actuator that compensates for both the free-air

resonance of the speaker and any dynamics from a pressure input.

The acoustic duct is a system that exhibits strong dynamics that when
coupled with the speaker system will cause large magnitude and phase
variations in the speaker response. These effects can then be minimized
through feed-back compensation. A numerical model is needed for the

acoustic duct before these effects can be demonstrated.

In this section, a model that accurately represents the duct pressure
response is developed. System equations are first presented, then they are
transformed into state space and transfer function representations. The
model is then verified by comparing it with experimental results obtained

from an acoustic duct.

20

21

System Model

An accurate system model of the acoustic duct is needed for modeling,
analysis. The linear second order wave equation modeling particle

displacement in a hard-walled, one-dimensional duct is [Seto 1971, Doak 1973]

 

82u(x,t) 32u(x,t) a 6(x)P(t) " a M,(t)
7-csz-5x—T —i§[5(x—x,)]§ pS (3.1)

where u(x,t) = particle displacement, c = wave speed (m / s), x = spatial

location (m), t = time (s), p = density of the medium (kg/ m3), Mi(t) = mass
flow input in the domain (kg/s), xi = location of mass flow input (m), S =
speaker area driving the mass flow input (m2), P(t) = pressure excitation at x =
0 (N/mz), and 5(x) = the Dirac delta function. The partially reflective

boundary condition at location x = L is the relationship between the spatial
gradient and the time gradient of the particle displacement and is expressed as

[Seto 1971, Spiekerman 1986]

gag) = -K[%]%tu-(L,t); K at 0 + 0i,1+ Oi,oo

(3.2)

where K = complex impedance of the termination end (dimensionless). The
duct end at x = 0 is modeled as a totally reflective end. This boundary

condition is

22

g—xu-(OJ) = O (3.3)

which corresponds to an open duct end. The acoustic pressure of the system

is related to the spatial gradient of the particle displacement by [Seto 1971]

P(x,t) = mag—(x1) (3.4)

The above four equations represent a mathematical model of the duct.

State Space Representation

To derive the state equations used throughout the analysis, separation of
variables is applied to the unforced version of (3.1), (3.2) and (3.3). Solving for

the separation constant and the eigenfunctions yields [Spiekerman 1990]

 

1n=—1-loge 1_K _nm, n=0,il,-_t2,... (3.5)
2L 1+ K L
¢,.(x) = e‘” + e"-‘ (3.6)

where A“ are the natural frequencies and ¢n(x) are the eigenfunctions of the

duct. For a duct with one mass flow rate as the input, the above equations can
be manipulated such that the following state space representation is

produced:

23

a(t) = A,

“C!

a(t) + B,

uct

m(t) (3.7)

where a(t) = the vector of modal wave amplitudes

Adm: the diagonal matrix [win]

 

Bduct = the matrix 1 d¢n(xi)
402.in6‘ (it

and m(t) = the mass flow input QM—

8t

The system output is the pressure at any position in the duct

P(xm,t) = Cducfaa) (3.8)

where P(xm,t)= the pressure in the duct at x = x,,,,

2 d¢n(xm)].

and Cm: the column vector [—pc dx

Equations (3.7) and (3.8) represent the state space formulation of the

acoustic duct with complex impedance, K, on the termination end.

Duct Transfer Function

A velocity to duct pressure transfer function can be computed from the

state space representation of the acoustic duct model for the case with one

24

mass flow input. The transfer function representation will be used for the

coupled speaker/ duct system model.

The duct transfer function can be computed numerically from

I’duct

m

 

Gduct (S) = = Cducr(SI - Aducr )4 B duct (3.9)

where 51 is the Laplace variable times an identity matrix and Gducr(S) is the
speaker velocity to duct pressure transfer function. The mass flow rate, m(t)
can be replaced by the speaker face velocity, vspkr’ by the relation,

m(t) = SDVWU), where SD is the speaker area. The transfer function, de(s),

will have a numerator which consists of a polynomial of order 2*n and a

denominator of order 2*n+1, where n is the number of modes in the model.

Duct Model Verification

The duct system model was verified through experimental testing. A
duct was assembled, and the system response was measured and compared
with the pressure predicted by the model. Figure 3.1 shows the experimental
setup with the duct, the driving speaker, the velocity transducer, and the

microphone.

25

 

Figure 3.1. Acoustic Duct Experimental Setup with Laser
Velocity Transducer and Microphone

An acoustic duct was constructed from 4-inch diameter PVC pipe. The
duct was 3.43 m long and was fitted with a tee joint so that a speaker could be
mounted in the duct wall, 0.56 m from the open end. A clear window [Hull
et. a1. 1993] was added to the tee joint so that the laser velocity transducer
could measure the velocity of the speaker face while maintaining pressure in

the duct, Figure 3.2. Both ends of the duct were left open.

 

Measured Duct “58:21:?
Sound Pressure Vglocity
Level Laser
Velocity
Transducer
PVC Pipe Open
W‘ / End
‘/Clear Indow
Fl I I

 

/ / \Tee Joint

Open Microphone
End

/ Speaker
Speaker Input
Signal

 

Figure 3.2. Experimental Acoustic Duct System

26

A half-inch Bruel 8: Kjaer microphone (model 4155) was placed in the
duct at the speaker location. The speaker velocity to duct pressure transfer
function was then measured with the signal analyzer with a frequency range
from 0 - 200 Hz. The speaker velocity to duct pressure transfer function was

also computed using the model. Values of 343 m/s and 1.21 kg/m3 were used
for the speed of sound, c, and the density of air, p. The value of the end

impedance was varied in the model until the model showed good agreement
with the measured response. An end impedance of 0.125+0j was determined
to give good model results. Figure 3.3 compares the measured and model

velocity to duct pressure transfer function.

 

 

 

 

 

 

 

 

 

O 4/ i i i ' -\i
o 50 100 150 200 250 300
Frequency (Hz)

8’

Q

8 :

N

S .

0' :

'2000 so 100 150 200 250 300

Frequency (Hz)

Figure 3.3. Comparison of Model and Measured Velocity to Duct Pressure
Transfer Function for Acoustic Duct

The measured response is shown with a solid line, and the model
response is shown with a dotted line. Acceptable agreement between the

model and the measured response was obtained.

27

These results verify that the model is accurate and can be used to model

the pressure input to the speaker system in the coupled speaker/ duct system.

4. COUPLED SPEAKER-DUCT SYSTEM

The coupled speaker/ duct system is presented in this chapter. The
coupled system is first modeled including the speaker velocity, the velocity
sensor, and the closed-loop feedback compensation. The model is then
verified by comparing it with experimental results obtained from the coupled

speaker / duct setup.

Modeling of Coupled Speaker-Duct System

In the previous discussions both the dynamics of a speaker and a duct
were modeled separately. The model of the speaker assumed that the
speaker face was exposed to atmospheric pressure. This implied that the
speaker velocity was only affected by the primary speaker voltage. The model

of the duct gave the pressure at a point in the duct given a velocity input.

These two systems can be coupled by allowing the velocity output of the
speaker to be the input to the duct and the pressure output of the duct to be
the input to the speaker. The velocity of the speaker face is then no longer
affected only by the primary speaker voltage but also by the pressure generated
in the duct, which must be determined from the coupled dynamics of the two

systems.

The model of the speaker dynamics is given by (2.1) and (2.3) rewritten

here in general form

28

29

xspkr(t)=A x (t)+B u (t)+B W (I) (4.1)

spkr spkr spkr spkr spkr spkr

yspkr(t) = Csperspkr(t) (4.2)

where usphls the speaker face pressure, Wm. 18 the speaker voltage, and yspkrlS

the speaker velocity. The model of the acoustic duct dynamics is given by

(3.9) rewritten here in the general form

iduct (t) = Aductxducr (t) + Bducruduct(t) (4.3)

yducta) = Cducrxducr (t) (4.4)
where umis the duct input velocity and yduais the duct pressure.

When the speaker face has constant pressure, the transfer function G(s)

can then be determined by dropping the term involving Pduct and solving for
Vspkr/Ep. For the coupled system, however, the output from the speaker, vspk,

becomes the input to the duct and the output to the duct, P becomes the input

to the speaker. Figure 4.1 illustrates the coupled system.

3O

 

 

 

 

WW“ I Speaker
SpFakCT .xspkr (t) = Axph xspkr (t) + Bspkruspkr (t) + Dspkr Wspkr(t)
Primary
VOltage yspkr( t) = Csperspkr(t)
Speaker
uducra) = yspkr(t) Face Speaker yducr (I) = uspkr (t)
Pressure VClOClty

 

Duct
.xducr (t) = Aducr xducr( t) + Bduct “duct (t)

y duct (t) = Cducrxduct(t)

 

 

 

Figure 4.1. Coupled Speaker-Duct System

The coupled system can be modeled by observing that y m = us”, and

“duct = yspkr'
(4.1) and (4.3) can be rewritten as

i....(t) = A....x....(t) + 3....C.,,..x.,...(t) (4.5)
55W“) = A.,.X.p..(t) + 3....Cd...xdm (t) + DWWWU) (4-6)

These can then be combined in matrix form as
12:11:22.. ”42:”"l(:::l+[uilw

The output to the system is the speaker velocity, given by

31

x ucr
yspkr/ducr(t) = [0 Cspkr ]{xd } (4‘8)

splcr

Equations (4.7) and (4.8) represent the state space equations for the

coupled speaker-duct system.

Velocity Sensor

The coupled system (4.7) and (4.8) can be used to model the response of
the speaker velocity and the velocity sensor presented in Chapter 2. The

speaker velocity, vspk, can be taken from (2.7) to be

Vrpkr(s) = Gvspkr / ep (S)Ep (S) + Gvspkr/ P(S)P(S) (49)

The velocity sensor response can be computed by combining the speaker
and duct transfer functions (2.7) and (3.9) and eliminating the pressure, P and
the secondary coil voltage, ebs terms. The velocity sensor to primary coil

voltage is given by

-Mcoi 3 GI MG»: krle Gduct
Viewer/Ep = bl I Gip/ep + 1P_G P PG +
(S+pl) ( vrpkr/P duct)

(4.10)
1 [ Gebs/ PGvspkr I edeuct ]
ab: / ep

— G
bl (1" Gvspkrl PGducr)

A complete derivation of (4.10) is given in Appendix F. Equations (4.9) and
(4.10) can be used to simulate the speaker velocity and the sensor velocity

response of the coupled system.

32

The feedback compensation strategy proposed in Chapter 2 can be applied
to the coupled system. The sensor velocity accounts for the pressure input as
well as the primary voltage input, so the closed-loop system should

compensate for the dynamics associated with both the speaker and the duct.

Coupled Speaker-Duct Model Verification

The coupled speaker/ duct system model was verified through
experimental testing. The speaker velocity model was first compared to
experimental results, then the velocity sensor was shown to accurately
predict the measured velocity. Finally, the velocity sensor was used in
feedback compensation. Figure 4.2 shows the coupled speaker / duct

experimental setup with the velocity transducer and the compensator.

 

Figure 4.2. Coupled Speaker/ Duct Experimental Setup with
Laser Velocity Transducer and Velocity Sensor

Speaker Velocity Model
The speaker/ duct system was setup as shown in Figure 3.2. The speaker
velocity to primary coil voltage transfer function was then measured with the

signal analyzer from 0 - 200 Hz. The model given by (4.9) was then used to

33

compute the model response. Figure 4.3 shows the model response compared
to the measured response. The model response is shown by the solid line and
the measured response is shown by the dashed line. An end impedance of

0.125+0j was used in the model.

Good agreement was obtained by the model. There is less than 5 dB

magnitude difference and 20 degrees and phase difference below 400 Hz.

 

  
  

   
 
 
   

 

 

 

 

 

 

 

10 I I I I I I I
o ' I
3-20 ....................................... z ........
c .
9 Mod 1
2 \ ,h 3.
81-30 . .‘ ..... ”L‘s ....... . ......
.1 F ' ‘4 'y'- !- ‘Measurhd‘a-Hq .
- - ~ . - - MK.“ ,‘
: : : : : : : K r
.40 1 1 1 1 1 1 1 M
50 100 150 200 250 300 350 400
Frequency(Hz)
100
a 50 ....... ....... g ........ g ....... g. ....... g ........
§ F ‘ 1" i. : : : : :
Oi Oi»... J. ................................. ' ........ ' ....... .(
.3; i \ l .
g-50L....E...\IE..“\: 3“. .‘ ....... : .......E. ....... : ........
_ ....“~.J. ..’ ..M0d8|...2 ...... .
a 100 : : : 5492371195?" --:~ ..... : "\
450 1 1 1 1 4 1 1
50 100 150 200 250 300 350 400
Frequency(Hz)

Figure 4.3. Comparison of Measured Frequency Response for
Coupled Duct-Speaker System with Model

The resonances of the duct can clearly be seen in the speaker velocity
response. These cause as large as 15 dB and 100 degrees of magnitude and
phase variation. The free-air resonance of the speaker is also superimposed
on the response. Clearly, the velocity of the speaker is affected by both the

speaker and the duct dynamics.

For the application of active noise control of an acoustic duct, the

objective is to attenuate the resonances in the duct. Figure 4.2 shows that the

34

speaker response has the most error exactly where the control effort is needed,
at the duct resonance frequencies. The response must be improved if the

speaker is to be an effective acoustic actuator.

Velocity Sensor

The velocity sensor presented in Chapter 2 was then applied to the
coupled speaker / duct system as show in Figure 4.4. A 2-inch foam plug was
placed in the termination end to add damping to the system. An end

impedance of 0.6+0.1j was used in the model.

Measured
Speaker
Velocity Laser
Velocity
Transducer

PVC Pipe Foam Plug

Tee Joint
\ /Clear Window /
I I 1 I
EV? Velocity <— desired velocity

 

 

 

 

 

 

Open Sensor ___.> sensor velocity
End /
Dual
Voice-Coil
Speaker

Figure 4.4. Diagram of Velocity Sensor in Speaker/ Duct System
Implementation

The sensor velocity to desired velocity and measured velocity to desired
velocity transfer functions were then measured from 0 - 200 Hz using the
signal analyzer. The modeled sensor velocity was also computed with (4.10)
using the value of p1 = 1000 Hz. Figure 4.5 shows good agreement between

these three signals.

35

 

 

 

 

 

 

     

m 0- .........................................................
'U

Q)

'U

.3 ' :

5, ---Sensor Measured g _

m : . : : :
2-401 ....... ......... g ......... g ......... g ......... .........

50 100 150 200 250 300 350
Frequency (Hz)

3100 ................................................
Q)
9.. .

m 0 ......... ......................................
g . . = . .
i -100 - ....... . . . . . . . ......... E' >YTN~-ma;;-_-_i._-‘-_-_-.-_-_-'i;;-,-.-u

 

 

 

50 100 150 200 250 300 350
Frequency (Hz)

Figure 4.5. Comparison of Measured, Modeled and Sensor Model Speaker
Velocity for Coupled Speaker/ Duct System

Figure 4.5 shows that the good agreement between the measured velocity
and the velocity sensor below approximately 200 Hz. There is less than 4 dB
magnitude and 20 degrees phase difference between the two responses. There
is significant phase error above 200 Hz which was seen with the experimental
results for the speaker only. Similarly, this error is attributed to the
inductance effects in the speaker which become significant at high

frequencies.

Velocity Feedback Compensation of Speaker

The velocity feedback compensation strategy presented in Chapter 2 was

then applied to the coupled speaker/ duct system. The proportional gain, KP,

36

was varied from 0 to 100; and the measured velocity to desired velocity
transfer function was measured from 0 - 200 Hz using the signal analyzer.
Figure 4.6 shows that the measured speaker velocity response approached the
desired velocity as the gain was increased. The noticeable deviation between
the velocity sensor and the measured velocity may contribute to the 45 degree

phase error above 100 Hz in the closed-loop system.

 

 

 

 

-4o 1 1 1 1 1 1 1 1 1
20 40 60 80 100 120 140 160 180 200
Frequency (Hz)

 

 

 

 

 

O I 0 l I O O O
.200 1 L 1 L 1 1 1 1

 

i
20 40 60 80 100 120 140 160 180 200
Frequency (Hz)

Figure 4.6. Comparison of Closed-Loop Speaker Velocity to Primary Speaker
Voltage Frequency Response for Open Loop and
Proportional Gains of 10, 20, and 100

The magnitude and phase variations exhibited in open-loop have been
minimized. The effect of the duct resonances and the free-air resonance of
the speaker are significantly reduced. With a value of Kp = 100 there is less
than 5 dB and 45 degrees magnitude and phase variation compared with 30
dB and 180 degrees in the uncompensated system. The compensated speaker
velocity is independent from the speaker and duct dynamics. This response is

ideal for an acoustic actuator.

5. CONCLUSIONS

Velocity feedback compensation for minimizing the magnitude and
phase variations in the velocity frequency response of a dual-voice coil
speaker coupled with an acoustic duct is presented in this work. The feedback
compensation uses a proportional controller to drive the speaker through the
primary speaker coil. The speaker cone velocity sensing is accomplished by a
new velocity sensor designed using the bond graph model of a speaker. It
uses a combination of speaker cone motion induced secondary coil voltage

and primary coil current.

The speaker velocity feedback compensation is experimentally shown to
reduce speaker velocity magnitude and phase variations. The compensated
speaker has variations less then 5 dB and 20 degrees in magnitude and phase
over a 0 - 200 Hz bandwidth compared to 30 dB and 180 degrees variation in
the uncompensated speaker. This advance makes the speaker an ideal
actuator in systems with strong coupling between the plant and the actuator,

as in active noise control of an acoustic duct.

37

APPENDIX

APPENDIX A

DERIVATION OF SPEAKER MODEL STATE EQUATIONS

The electromagnetic speaker is a system involving mechanical,
electrical and magnetic energy domains. This class of problem is well suited
to modeling using the Bond Graph technique [Karnopp & Rosenberg 1983].
Figure A.1 shows the bond graph diagram of the dual voice-coil speaker

system.
Rcoil

Secondary Coil «P

SM 1 M (:bY Ispkr

Icoil 1 jib TFI—A

1!):ch / ESRSPHQ
Primary Coil bl
S __>|l HA GY
1/
Rm
Rcoil
Figure A.1. Bond Graph of Dual Voice-Coil Speaker System

The Bond Graph arrows represent power flow from the electrical

voltage, 5,, applied at the primary coil through the model’s elements. Power

38

39

is dissipated in resistive elements Rm”, R... and RM, 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 ’C’ elements. The open circuit on
the secondary coil is represented by a secondary coil current, is =0 imposed by
the source of flow, 5,.

The energy variables in the bond graph model include coil flux

linkages, 11p and 1,, speaker cone displacement 1cm, and speaker cone velocity,
vW. A linear approximation is used to relate the flux linkages, 71p and AS, of
the multiport 1.0.1 field to the port currents i, and i, in the primary and

secondary coils.

A = Ii (A.1)
M

coil . p
I

coil .3

I .
where, I =[ “'1’”

i
M ] is the inductance matrix, i = (3" ] is the vector of

l

coil .3 s

and I in

oil. p coil.s

A

S

A
port currents and A. =( P ] is the vector of coil flux linkages. I,

the inductance matrix are the self inductances of the primary and secondary

coils, and M and M

coil , p coil .3

are the mutual inductances between primary and
secondary coils. The self and mutual inductance terms in the inductance
matrix are equal for audio voice-coil speakers because the coils are designed to

be identical in construction.

Mcoil.p = Mcoil.s = Mcoil (A°2)
Icoil.p = Ic011,: = Icoil (A3)

For this application, the secondary coil is always an open circuit. Therefore

the current in the secondary coil, i,(t)=0. With this, the relation between flux
linkages, 11p and II, and primary current, i,,(t) is given by

4O

A=I

p coil 1 p

mp

A=M (AS

coil 1 p

The state equations for the speaker model can be written from the bond
graph in state space form by choosing energy variables as state variables and

appropriate input and output variables as

 

 

 

 

 

 

xm=gpm+gwwm (Aw
where
" 0 1 0 I
-l —RSP,,, bl 0 0
— —S
Aspkr = CSPk’ISPkT Ispkr (Ispkrlcoil) I Bspkr = 0 T—Q' (A.7)
splrr
l 0
0 —b1 -(Rcoil + Rm)
__ Icoil ..
xw
The state variables are 15.7.1. = ”so , speaker face displacement xspkr, speaker

A

P

face velocity, vspkr and primary coil flux AP, and the input variables are

e
“3,11, = [ Pf , speaker primary coil voltage, ep, and speaker face pressure, P.

The output variables can be arranged in state equation format as,

 

 

yspkr = Csperspkr + D spkruspkr (A8)
where
' M. M. ‘ _ _
O bl[l -—‘fl) -1—2ML(Rcoil + Rm) Mc O
coil coil 1 [c
cw=o o 1. ,QW=O 0 (A%
coil 0 O

 

 

O 1 0 ' ‘

 

 

41

eb

S

The output variables, 337.11 = in are speaker secondary coil voltage, e,,(t),

v3?”

speaker drive current in the primary coil, i,(t), and speaker face velocity, vspkr.
The voltage, e,S(t), introduced in the secondary speaker coil is obtained
from the bond graph model by summing the voltages across the 1—junction
on the secondary coil side of the Icoil element. These voltages include the
voltage, ebs,(t), due to the mechanical motion of the speaker cone and the

voltage, eb,2(t) due to the mutual inductance, M between the coils.

coil I

ebs(t) = 9131(01' eb32(t) (A.10)

The voltage, ebs,(t), introduced due to the mechanical motion of the speaker

cone is linearly proportional to the speaker cone velocity, v.11.

ebsl (t) = bl vspkr (t) (A'll)

The voltage, em(t), in the secondary speaker coil can be written from the
constitutive relation of the coils by noting that the secondary coil current, is:

0 due to the open circuit condition on the secondary coil.

eb32(t) = A.11 = Mc M

. A.12
orl dt ( )

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

 

 

 

 

di (t)

eb,(t) = bl vspk, + Mm" 4;?— (A.13)

The primary coil current, i,,(t) can be replaced by Ap /Icoil yielding

Mcoi d1 (t)

e,”(t) = bl vspk, + EL—jr (A.14)

d)» (t) . . .
The term 6”” can be replaced With row 2 from (A.7) givmg,
eb,(t) = bl[1- AI'I‘ )v,(t) — 1:1; (R, + Rm)Ap(t) + All‘ ep(t) (A.15)

42

The output equation can now be written for the output variables,

secondary coil voltage, eb,(t), primary coil current, i,,(t) and speaker velocity,

vspk, (t) in terms of the state variables, speaker displacement, x3 speaker

pkr’

velocity, V, and primary coil flux linkage, AP, and the input variables,

pkr I

primary coil voltage, ep, and pressure, P as,

 

 

 

 

0 “(l—fl] ——M—2“1(Rcm., + Rm) "MC ‘
eb coil coil {x kr I O
s 1 sp c [e
ip = O O vspk, + 0 O p) (A.16)
Icoil KP
vspkr \ AP 0 O
_o 1 0 _ ‘ ‘

 

 

Equations (A.7) and (A.16) define the model of a dual voice-coil speaker.

APPENDIX B

SPEAKER PARAMETER IDENTIFICATION

The physical parameters of the speaker were determined using a

methodology given by [Radcliffe, Gogate, 1992].

1. The speaker compliance, C k, is given by staticly loading the
speaker face with a digital sforce meter and measuring
displacement and determining slope from force vs. displacement

graph.

Table B.1. Speaker Compliance Data

 

 

 

 

 

 

 

Displacement Force
(m) (N)

0 0
0.002 3.1392
0.004 7.7499
0.006 14.96025

 

 

43

3.

Force (N)
d—L—t—L
omeoacoorosso:

 

0 0.001 0.002 0.003 0.004 0.005 0.006
Displacement (m)

Figure 3.1. Force Vs Displacement Graph for Speaker

Cspkr = 0.000868 m/ N25

The speaker resonant frequency f“. is determined from the
frequency response of speaker velocity to primary coil voltage,
Figure B.2.

 

0 V Y I v 1 fi" 1 f V v V v T v V v
o o o e e I u e I n c n n o u e a

c o o I I e I o I n 1 o o e s u

o o o o c I I n u e o v o I u o -

o I c c o I o c o . e e o n a '

Gain dB

 

 

 

 

 

 

 

 

1 0
Frequency ( Hz)

Figure 8.2. Frequency Response of Speaker Face Velocity to Primary Coil
Voltage

fn = 62 Hz

The damping factor Q is determined from the half power points
on Figure B2 and the quality factor, Q.

resonance frequency f“: 62 Hz

45

half power frequencies f1 = 36 Hz, 125 Hz
Quality Factor, Q is given by,

 

=—f;'——= 62 = -
Q fz-fl 125-36 0.697 (B 1)

Damping ratio C is given by,

€=E=§Ea6—96—)=0.718 (3'2)

The primary and secondary coil resistance is obtained by
measuring each resistance with an ohm meter.

Rcoil = 4.875 ohms

The speaker inertia, Ispkr is determined from

1 1
I = =
1’” Cwaygpz (0.000868m/N)(389rad/s)2

 

 

= 0.0076kg (B-3)

The electromagnetic coupling factor, bl is obtained from
Ispkchoil [(0)3 _ wlz)2 + 4(Cwlwn )2]

flail ((02 — (of )]2 +[2Q’cofa1u]2

where M is the magnitude of the frequency response at a
frequency (1).. Using M=-20 dB and (1)“: 251.33 rad / s,

 

bl = 10”“2‘” (13-4)

bl = 2.4566 N/ A

The equivalent speaker area is given by 87% of the advertised
cone area as,

2
SD = 0.87% D) = 0.87(0.0153m2) = 0.0133»:2 (13-5)

where the advertised diameter, D is 0.140m (5.5 inches).

The speaker viscous friction constant, Rspkr is given by

I 2
W" i — Pi- = 3.745 N sec/ m (B-6)

R =
spkr
Cspkr Q Rcoil

 

46

A summary of the speaker parameters is given blow.

Table 3.2. Speaker System Physical Parameters

 

 

 

 

 

 

 

Parameter Symbol Value
mechanical inertia Ispkr 0.0076 Kg
mechanical compliance Cspkr 0.000868 m / N
viscous friction Rspkr 3.745 N sec / m
electromagnetic coupling factor bl 2.45 N/ A
coil resistance Rm“ 4.875 ohm
equivalent speaker area SD 001331112

 

 

 

 

 

APPENDIX C

DERIVATION OF SPEAKER MODEL TRANSFER FUNCTIONS

The transfer function matrix for the speaker model can be determined
from (2.1) and (2.3). The speaker model includes two inputs and three
outputs, resulting in a transfer function matrix of dimension 3x2. The

steady-state output of the system is given by [Rough 1996]

Y(s) = [C(sI — A)" B + D]U(s) (C1)

where A is the state matrix, B is the input matrix, C is the output matrix, D is
the direct feed-through matrix, ’sI’ is the Laplace variable multiplied by an
identity matrix, U(s) and Y(s) is are the input and output vectors in the
Laplace domain.

The transfer function matrix is defined by
G(s) = C(sI — A)" B + D (C2)

The state equation matrices for the speaker model are

 

 

 

 

 

 

 

 

' 0 1 0 1
—1 _Rspkr bl 0 g
Aspk' = C5”, I‘m," ISP’" (Ispkr, coil) 1 Bspkr = 0 ; D
spkr
.1 0 _
0 -bl _(Rcoil + Rm)
- Icoil _

47

48

 

 

 

 

 

 

 

0 m(l—Mfl] _fl2€L”-(Rwfl + Rm) "MC 0‘
coil coil
I
1 c
Cspkr = O O 7— ' Dspkr = O O
corl O O-J
_0 1 0 j ‘
Computing (sI-A) gives
F s —l 0 q
R _
(SI—A): s+ W” bl
Cspkr Ispkr Ispkr ( Ispkr I coil )
O bl S+fl
__ Icoil _

 

 

Computing C(sI-A)‘1B yields

ebs(s) Gebs/ep Gebs/P [e (3))
P

ip(s) = Gip/(p ip/P P(S)
vspkr (S) Gvspltr/ cp Gvspkr/ P

where each element in the transfer function matrix G(s) is given by

G = [( CWk'ISPkacoil )S3 + (CspkrRspkrM coil )sz + (blzcspkr + M coil )‘fl/
den

cbs I cp

cp/cp '—

(bl C b)s
Gvspkr / cp = [ SP %dcn

["bl CSPkISDkIcoiI _ Alcoil)s2 + RCOHSH/
Gebs/P = Gm

GiP/P = [(bl 80C?” )%cn
Gvspkrl P = {-SDC‘W‘JICOHSZ + Rem-13%“

and the denominator of the G(s) matrix is given as,

G _ [(Cspkrlspkr )SZ + (Csprspk,)S + l]
Gden

(C3)

(C4)

(C5)

(C6)

(C7)

(C8)

(C9)

(C10)

49

G,,,,=(C I Io,,)s3+(C I R +C I R )s2+

spkr spkr c spkr spkr coil spkr coil spkr

(bIZCW + I + CWR R h)s + R,

coil coil sp

(C.11)

oil

APPENDIX D

COMPENSATOR CIRCUIT IMPLEMENTATION

The speaker compensation is implemented with an analog circuit
which includes the velocity sensor and the proportional controller.

The velocity sensor equation is given by,

Vsp1.(5) = ”1.511(5) — Hp(s)Ip(s) (D.1)

where Hbs = 1/ bl and HP(s)= sMcon/ bl.

The secondary coil voltage, e,, can be measured directly from the
speaker coil, the primary coil current, i, can be determined from the voltage
across the resistor Rn". Hbs is a pure gain (1 / bl), and the term Hp(s) is a
differentiator transfer function because it contains an ’s’ in the numerator.

Such a transfer function can not be realized exactly but an approximation

Hp(s) can be used, where

 

A = coil 5'
Hp(s) bl [s + P1] (D.2)

where p1 is a pole location selected such that Hp(s) approximates Hp(s) over
the controller bandwidth.

Feedback compensation can now be implemented using the signal
from the velocity sensor to compute the error between the desired velocity

and the sensor velocity and a proportional controller to drive the speaker

50

51

velocity to the desired velocity. The proportional controller in block diagram

form is given in Figure D.1.

 

 

 

 

 

 

 

 

 

 

 

 

vd (t) Proportional e(t) Speak? Vspkr“)
Controller -—> Dynamxcs __>
+ Kp Gspkr
Velocity
Sensor
H(S)

 

 

 

Figure D.1. Proportional Feedback Controller
A circuit diagram of the compensation circuit used in the experimental

portion of this study is given here.
from speaker
primary coil
@ 100k 10" 10k

10k 10k

 

1k

10k 1“
It) 9
:- — 1 k lino‘ltuatge 1"

 

 

to amplifier
1k and speaker
® , . primary coil
from speaker — _
secondary coil - ': —

Figure D.2. Speaker Compensation Circuit Diagram
An external amplifier was used to implement the gain KP. Figure D.3

shows the overall arrangement of the compensated speaker components.

52

Input
voyage 6p 8 eaker
d ‘ primary p

Cempensator :3. Amplifier] SpeakercoL

Kp l 0
¢

secondary
speakercofl

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure D.3. Overall Arrangement of Compensated Speaker Components

APPENDIX E

ACOUSTIC DUCT STATE EQUATION COMPUTATION ALGORITHM

The acoustic duct state equations were computed from the equations
given in Section 3 using a numerical algorithm in the Matlab software
programming environment. The algorithm, given blow takes the number of
modes to be generated and returns the state equation matrices, A, B, C, and D.

11....- rou- 0. 011--1_: ‘ 0-1' '4. t-_°Ec.--.101.

function [A,B,C,D,N]=genstsp(N)
%M-Pile to generate state space matricies for acoustic duct
%Input number of modes N to include in model

%Define Variables

L=140*.0254; %Length of Duct 2.1336m,118in
K=0.125+0*i; %Impedance of Duct End 0.1268+0.0*i
S=0.009;%.25*pi*(6*.0254)"2 %Speaker area m"2

xm=22*.0254; %Location of measurement in duct 0.7874
xi=22*.0254; %Location of disturbance input

c=343; %Wave speed 343 m/s

ro=1.21; %Density of medium air 1.21 kg/m"3

for n: -N:N

count=n+N+1;
Ln(count)=1/2/L*log((1-K)/(1+K))-n*pi*i/L;
pdotm(count)=Ln(count)*(exp(Ln(count)*xm)-exp(-Ln(count)*xm));
pdoti(count)=Ln(count)*(exp(Ln(count)*xi)-exp(-Ln(count)*xi));
B(count,1)=pdoti(count)/(4*c*(Ln(count)"2)*L‘ro*S); %controller
Dn(count,1)=-1/(2*c*L*r0*Ln(count)); %disturbance
C(count,1)=-ro*c*c*pdotm(count);

end

A=diag(c*Ln);
D=[0];
end

53

APPENDIX F

COUPLED SPEAKER/DUCT VELOCITY SENSOR TRANSFER
FUNCTION

The coupled system can be used to model the response of the speaker

velocity and the velocity sensor presented in Chapter 2. The speaker velocity,

Vspkr is given by
Eb! (s) Gebsl ep Gcbs I P E (S)
IP(s) = GiP/ep Gip/P [£250] (F°1)
‘lspkr (S) Gvspkr I ep Gvspkr I P
as
Vspkr (S) = Gvspkr/ ep (S)Ep (S) + Gvspkrl P(S)P(s) (F.2)

The velocity sensor response can be computed by combining the speaker and
duct transfer functions and eliminating the pressure, P and secondary coil

voltage, Ebs. The duct pressure to speaker velocity transfer function is given

by

Pduct = G
duct
duct

(F3)

The pressure can be eliminate from (F2) by substituting (F.3) which gives

Vspkr(s) = Gvspkr/ cp (S)Ep (S) + Gvspkr/ P (S)Gduct (s)‘/spkr (S) (F'4)

54

 

55

which can be solved for the transfer function of speaker velocity to primary

speaker voltage as

 

‘lsp% = Gvspkr / ep ( F . 5)
P (1 - Gvspkrl PGduct)
The secondary speaker voltage was given by (F .1). The pressure, P can

be eliminated from (El) using (F.3), giving

Ebs(s)=Ge EP+GWPG v (F.6)

bs / ep due: spkr

The velocity can be eliminated from (E6) using (F5) giving the

secondary speaker voltage to primary speaker voltage transfer function as,

Gab: I PGvspkr I ep Gduct
(1 - vapkrl PGduct)

 

EbS/EP=G +

eh: I ep

(F7)

The primary speaker current, IP is given by (RI). The pressure and velocity

can be eliminated as before, giving,

 

G. G G
1}) /Ep = Gip/ep + lp/P vspkr/ep duct (F8)
(1" Gvspkr/ PGduct)

The velocity sensor equation is given by
vspk,(s) = HbsEb,(s) - Hp (3)11, (3) (F9)

where Hbs = 1/ bl and HP(s)= sMcofl/ bl. (F.5) and (RS) can be substituted into
(R9) to give the sensor velocity to primary speaker voltage transfer function

 

V / E -_:flfl‘:_ G + GiP/PGvspkr/edeua +
sensor p - bl iplep 1-G G
(3+ pl) ( vspkr/P duct)

_1_ b I + Gebs/ PGvspkr/ ep Gduct
bl G 3 GP (1 - Gvspkrl PGduct)

(F.10)

These equations, (F2) and (RN) can be used to simulate the speaker velocity

and the sensor velocity response of the coupled system.

 

LIST OF REFERENCES

LIST OF REFERENCES

Bradley, P., ”Active Assault on Cabin Noise”, Commercial Aviation vol
77, September, 1995, 6 pp.

Warner, J., ”Active Noise Control in an Ofl-Road Vehicle Cab” Noise and
Vibrations Worldwide, vol 26 n, 7 July, 1995.

General Motors 1994, Report #.

Hull A. J., Radcliffe C. J. Southward S. C., ”Global Active Noise Control of
a One-Dimensional Acoustic Duct Using a Feedback Controller” Journal of

Dynamic Systems, Measurement, and Control, vol 115, September, 1993.

Radcliffe C. J., Gogate, S. D., ”Velocity Feedback Compensation of
Electromechanical Speakers for Acoustic Applications”, International
Federation of Automatic Control, Triennial World Congress, July, 1996.

Karnopp, D. C., D. L. Margolis, and R. C. Rosenberg, ”System Dynamics: A
Unified Approach”, New York, John Wiley 8: Sons, Inc., 1990.

IEEE Standard 219-1975, IEEE Standard Committee of Acoustics, Speech,
and Signal Processing Group, ”IEEE Recommended Practice for Loudspeaker
Measurements”, IEEE std. 219-1975.

Rough, W. 1., ”Linear System Theory”, New Jersey, Prentice-Hall, Inc.
1996.

Radcliffe, C. J., Gogate, S. D., ”Identification and Modeling Speaker
Dynamics for Acoustic Control Applications”, ASME Symposium on Active
Control of Noise and Vibration, 1992.

Seto, W. W., ”Theory and Problems of Acoustics”, New York, McGraw-
Hill Book Company, 1971.

Doak, P. E., ”Excitation, Transmission and Radiation of Sound From
Source Distributions in Hard-Walled Ducts of Finite Length (I): The Effects of
Duct Cross-Section Geometry and Source Distributions Space-Time Pattern,”
Journal of Sound and Vibration, v 31, n 1 pp1-72, 1973.

Spiekerman, C. E. and Radcliffe, C. J., ”One—Dimensional Acoustic
Response with Mixed Boundary Condition: Separating Total Response into
Propagating and Standing Wave Components,”: Doctoral Dissertation,
Michigan State University, 1986.

56

57

Hull A. J., Radcliffe C. J., ”State Space Representation of the Nonself-
Adjoint Acoustic Duct System,” Journal of Vibration and Acoustics v 112,
October, 1990.

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