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ACTIVE CONTROL OF SOUND
Professor Mike Brennan
Institute of Sound and Vibration ResearchUniversity of Southampton, UK
Active control of sound
• Active control of sound in ducts Single secondary source Two secondary sources Where does the power go? Control of harmonic disturbances Control of random disturbances Single channel feedforward control Constraint of Causality
• Active control of sound in enclosures Cars Aircraft Active head sets Vibroacoustic control
Passive Control of Sound
Passive control relies on barriers, absorption and damping.
It works well when the acoustic wavelength is short compared with typical dimensions Higher frequency solution.
Sound source Observer
Active Control of Sound
Acoustic or structural actuators are driven to cancel waves:
It works well when the acoustic wavelength is long compared with typical dimensions Lower frequency solution.
Observer
Sound source
Patent for Active Control of sound by Paul Lueg
1936
Active Control of Duct-Borne Sound
Loudspeaker source in a duct
If the frequency of interest is such that the acoustic wavelength is greaterthan twice the dust cross-section then it can be modelled as a pair of massless pistons forced to oscillate apart with a fluctuating volume velocity q(t) between them.
Loudspeaker source in a duct
For x > 0 the complex pressure and particle velocity fluctuations can be written as:
( )q t
x
( )
( )
jkxo o
jkx
p x c U e
u x U e
0x
For x < 0 ( )
( )
jkxo o
jkx
p x c U e
u x U e
0x
where U+ is the velocity of the right-hand piston and U- is the velocity of the left-hand piston.
The plane monopole source
x
( , )p x t (0) o o o op c U c U
x
( , )u x tU U
x
The plane monopole source
x
q
2q SU
We define the source strength as
( )2
jkxo ocp x q e
S
So0x
( )2
jkxo ocp x q e
S
0x
area of cross-section S
Choose a secondary source strength to set pressure field downstream of secondary source to zero
Cancellation of downstream radiation using a single secondary source
0x
pq sq
Primary source Secondary source
The fields due to the primary and secondary source are
( )2
jk xo op p
cp x q e
S
( )2
jk x Lo os s
cp x q e
S
Use the principle of superposition to calculate the net sound field
( ) ( ) ( )p sp x p x p x
x L
Cancellation of downstream radiation
0x
( )pq t ( )sq tPrimary source Secondary source
x L
( ) 0, p x x L This requirement is
which leads to ( ) 0,
2 2jk x Ljkxo o o o
p p s
c cp x q e q e x L
S S
Thus jkLs pq q e
that is the secondary source is a delayed inverted form of the primary source.
Noting that the Fourier transform pair , we
can write ( ) ( )
oj to
s p o
x t t Xe
q t q t L c
The net sound field in the duct
The field between the primary and secondary sources is give by
( ) , 02
jk L x jk L xjkLo op
cp x q e e e x L
S
Upstream of the primary source it is given by
2( ) 1 , 02
jkx j kLo op
cp x q e e x
S
Downstream of the secondary source it is given by
( ) 0, p x x L
The net sound field in the duct
-1 0 10
0.5
1
1.5
2
2.5
-1 0 10
0.5
1
1.5
2
2.5
-1 0 10
0.5
1
1.5
2
2.5
-1 0 10
0.5
1
1.5
2
2.5
( )
2o o
p
p x
cq
S
x
2L 5 8L
3 4L L
Note that when L=nλ/2 the pressure upstream of the primary source =0
Time domain interpretation
0x
( )pq t ( )sq t
Primary source Secondary source
x L
( )s po
Lq t q t
c
( , )p x t
x
Cancellation of downstream radiation using a pair of sources
0x
( )pq t ( )sq tPrimary source Secondary sources
x L
2( )sq t
x L d
With two sources it is possible to ensure zero radiation upstream of the Secondary source pair by setting
1 2jkd
s sq q e
Downstream of the second secondary source the net pressure field can be set to zero by setting
2 2sin
jkLp
s
q eq
j kd
The net sound field in the duct
The field upstream of the secondary sources is given by
( ) , 2
jk xo op
cp x q e x L
S
Between the secondary sources it is given by
( ) , 4 sin
jk x L d jk x L djkLo op
cp x q e e e L x L d
j S kd
Downstream of the secondary sources it is given by
( ) 0, p x x L d
0 0.5 1 1.5 2 2.50
1
2
3
0 0.5 1 1.5 2 2.50
1
2
3
0 0.5 1 1.5 2 2.50
1
2
3
0 0.5 1 1.5 2 2.50
1
2
3
The net sound field in the duct
( )
2o o
p
p x
cq
S
x
9 16d 5 8d
3 4d 15 16d
Time domain interpretation
1 2jkd
s sq q e
The secondary sources are given by
2 2sin
jkLp
s
q eq
j kd
To enable interpretation in the time domain let us choose a primary source strength whose Fourier transform is some function i.e.,
( ) 2sinF j kd
( ) 2sin ( ) ( ) ( )jkd jkdpQ j kd F e e F
In the time domain this assumes
( )p o oq t f t d c f t d c
It then follows that 2( ) ( ) jkLsQ F e
2( )s oq t f t L c
1 2( )s s oq t q t d c
or in the time domain
So
Time domain interpretation
0x
( )pq t 1( )sq t
Primary source Secondary sources
x L
2( )sq t
x L d
2( )so
Lq t f t
c
1 2( )s s
o
dq t q t
c
( )p
o o
d dq t f t f t
c c
( , )p x t
x
Sound absorption by real sources
i
upm
k c
R
2 2 21Re
2 e me aZi Z u Zu Electrical power supplied
Electricalimpedance
Mechanical impedance
Acousticalimpedance
The acoustical power can be negative; in such cases less electrical power will be required to sustain a given piston velocity u
The influence of reflections from the primary source
0x
( )pq t ( )sq tPrimary source Secondary source
x L
To set the pressure downstream of the secondary source to zero
jkD jkDp
s jkL jkL
q e R eq
e R e
Absorbing surface havinga complex reflectioncoefficient R
x D
The influence of reflections from the primary source
0x
( )pq t ( )sq tPrimary source Secondary source
x L
For a primary source next to the reflecting surface (D=0)
1ps jkL jkL
q Rq
e R e
Now, if R=1, then
cosp
s
kL
Thus the secondary source strength required to cancel the sound fieldbecomes infinite when
3 5, , ........
4 4 4L
Adaptation in Feedforward Control
An error microphone is introduced to monitor the performance.
Changes in the disturbance and plant response, from loudspeaker to the microphone, require adaptation of the feedforward controller.
TRANSFORMER
AMPLIFIER PHASEANGLE
AMPLI-TUDE
HARMONIC SOURCE
SOUND ANALYZER
SOUND LEVEL METER
LOUDSPEAKER MICROPHONE
Active Control of Transformer Noise, Conover 1956
Single channel feedforward control
PeriodicPrimary source
( )G j( )x t ( )y t
( )e t
Electricalreference signal
Electroniccontroller
Secondary source
Errorsensor
(Unaffected by secondary source)
( )G j ( )C j ( )X ( )Y
( )D
( )E
Reference signal Electronic
controllerElectroacousticsystem
Errorsignal
Primarycontribution
Single channel feedforward control
( )G j ( )C j ( )X ( )Y
( )D
( )E
Reference signal Electronic
controllerElectroacousticsystem
Errorsignal
Primarycontribution
E D G j C j X
At the n-th harmonic the error signal can be completely cancelled if
on
oo
o
D nG jn
C jn
( ) ojn tx t e
Reference signal is
Control of random noise in a duct
( )G j( )x t ( )y t
( )e t
Detection sensor Electronic
controller
Secondary source
Errorsensor
Sound fromPrimary source
1. The detected signal x(t) is generally influenced by the electroacoustics of the feedback path
2. There is a constraint of causality on the controller
There are two main differences between the control of random andharmonic disturbances
Control of random noise in a duct
( )G j ( )C j ( )X ( )Y
( )D
( )E
Signal due to primary source
Controller
Error pathErrorsignal
Primary path
( )U
( )P j
( )F j1( )N
2( )N
( )S
Measurement noiseat detection sensor
Measurement noiseat detection sensor
Feedback path
( ) ( )( )
( ) 1 ( ) ( )
Y G jH j
U G j F j
Signal to secondary source
Signal atdetection sensor
1( ) ( ) ( )U S N 2( ) ( ) ( )D P j S N
Optimal controller
( )H j ( )C j ( )U ( )Y
( )D
( )E
Primary and measurementnoise
Controller andfeedback path
Error path Errorsignal
disturbance and measurement noise
The block diagram becomes
Since the system is linear and time-invariant, we can transposethe signal paths to give
( )H j( )C j ( )U ( )R
( )D
( )E
Controller andfeedback path
Error path Errorsignal
Filtered reference signal
( ) ( ) ( ) ( )E D H j R where ( ) ( ) ( )R C j U Filtered reference signal
Optimal controllerPower spectral density of the error signal is
*EeeS E E where E[ ] is the expectation operator and * denote complex conjugation
Now E D H j R
So * * *ee dd rd rd rrS S S H H j S H j S H j
This can be written in standard Hermitian quadratic form as (dropping the explicit dependence on frequency)
* * *eeS H AH H b b H c
where , , rr rd ddA S b S c S
Optimal controllerThe power spectral density of the error signal can be written as
eeS
Re H Im H
2 2Re Im 2Re Re 2Im ImeeS A H H b H b H c
opt
Re H opt
Im H
Global minimum
Set derivatives of with respect to
the Re and Im to zero
which leads to
eeS
H H
1
optRe ReH A b
1
optIm ImH A b
opt opt optRe ImH H j H
1optH A bSo
Optimal controller
To find minimum error substitute into1optH A b
* * *eeS H AH H b b H c
To give * 1(min)eeS c b A b
which can be written as
2
(min)rd
ee ddrr
SS S
S
Now2
rr uuS C S and2 2 2
rd udS C S
Coherence between signals fromdetection sensor and error sensorprior to control 2
ud
2
(min) 1ee ud
dd dd uu
S S
S S S So
The maximum possible attenuation in dB at each frequency is thus given by
210Attenuation in dB 10log 1 ud
( )H j( )C j ( )U ( )R
( )D
( )E
Optimal controller
( )G j ( )C j ( )X
( )D
( )E Controller
Error pathErrorsignal
( )U
( )F j
Feedback path ( ) ( )( )
( ) 1 ( ) ( )
Y G jH j
U G j F j
optopt
opt
( )( )
1 ( ) ( )
H jG j
H j F j
So the optimal controller is given by
1optbut ( ) rd ud
rr uu
S SH j A b
S CS
optSo ( ) ud
uu ud
SG j
CS FS
Digital implementation of the controller
( )G j( )x t ( )y t
( )e t
Detection sensor Electronic
controller
Secondary source
Errorsensor
Sound fromPrimary source
L
( )x t( )y t
Digital filterAnalogueto digitalconverter
ADC
DAC
Digital toanalogueconverter
( )x n ( )y n
Analogueanti aliasconverter
Analoguereconstructionfilter
( )DG z
Digital implementation of the controller
The overall frequency response of the controller is
( ) ( ) j TDAG j eG Gj
Frequency response of filters and data converters Digital filter
Sampling time
The controller must have a delay of seconds oL c
Causality condition
Approximate delay through an analogue filter is roughly due to 45°phase lag or 1/8 cycle of delay at its cut-off frequency, fc
Total delay through two filters which have a total of n poles is n/8fc
The cut-off frequency is typically 1/3 the sampling frequency (fs=1/T), so that fc=fs/3=1/(3T)
Allowing 1 sample delay for the data converters and the digital filtermeans the total delay is given by
1 3 8A T n
Rectangular duct with largest dimension D=0.5m – single channel control can only be achieved below about 300 Hz
Causality condition - example
Sampling frequency = 1kHz (T=1ms)
Two 4th order analogue filters (n=8)
Delay in analogue path is about 4ms
( )G j( )x t ( )y t
( )e t
Detection sensor Electronic
controller
Secondary source
Errorsensor
Sound fromPrimary source
L
D
1.5, which is about three times the width of the ductA oL c
Active control of sound in a duct – experimental work (Roure 1985)
Side view
Plan view
Active control of sound in a duct – experimental work (Roure 1985)
Frequency (Hz)
dB
Amplitude spectra of the fan noise at the error microphone with a meanduct velocity of 9m/s
Active control off
Active control on
Active control of sound in enclosures
Electronic Sound Absorber
H.F. Olson and E.G. May, Journal of the Acoustical Society of America,pp. 1130-1136, 1953
Active Control of Sound inside Cars
Low-frequency engine noise in the car cabin can be controlled with 4 loudspeakers, also used for audio, and 8 microphones, also used for hands-free communication (Elliott et al. 1986).
Initial Demonstration Vehicle
Measured Results in a Demonstration Vehicle
A-weighted sound pressure level at engine firing frequency
Active Sound Control in Propeller Aircraft
System is standard fit on Dash 8 Q400 (Stothers et al. 2002)
Active Sound Control in Propeller Aircraft
www.bombardier.com
Periodic excitation generates intense harmonic soundfield inside cabin
Active Sound Control in Propeller Aircraft
Spectrum of Pressure Inside Propeller Aircraft
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0-4 5
-4 0
-3 5
-3 0
-2 5
-2 0
-1 5
-1 0
-5
0
5
F re q u e n c y (H z )
dB
(A)
re a
rbit
rary
le
ve
l
D A S H -8 S e r ie s 2 0 0 : R e d u c t io n 1 1 . 3 d B (L ) , 8 . 2 d B (A )Dash-8 Series 200: Reduction 11.3 dB(L), 8.2 dB(A)
Frequency (Hz)
dB
(A)
re a
rbit
rary
le
ve
l
Active Sound Control in Propeller Aircraft
Centralised digital system made by Ultra Electronics controls 5 harmonics with 48 structural actuators at 72 acoustic sensors, distributed throughout cabin.
Control System for Propeller Aircraft Active Noise System
Active Sound Control in Propeller Aircraft
Typical Performance of an Active Aircraft System
Single multichannel centralised digital controller used with 48 actuators and 72 sensors distributed throughout the cabin
SYSTEM OFF
SYSTEM ON
Feedback control of Sound
Active Headset using Feedback Control
– H
Earshell
Cushion
Secondary loudspeaker
Analogue controller
Error microphone
Enclosed volume, V
Ear
If no external reference signal is available, conventional feedback control can be used to control sound at low frequencies.
Feedback control of Sound
Active Headset using Feedback Control
Frequency (Hz)
dB Active control off
Active control on
Feedback control of Sound
Active Headset using Feedback Control
www.Bose.com
Active headrest
Active headrest – zones of quiet
kL=0.2 KL=0.5
KL=1 KL=2
x
L
10dB
20dB
Active Vibroacoustic Control
The Problem
Transmitted sound power
Incident sound power
Simply supported panel
baffle
Objective: To minimise the transmitted sound power
The Active Control System
Accelerometer
Piezoceramic actuator
Panel
Analogue controller
Piezoceramic Actuators
F F
d d
plate
actuator
M M
plate
F F
Active Control Performance (simulations)
Frequency (Hz)
Sou
nd t
rans
mis
sion
rat
io (
dB)
Increasing gain
Feedback gain
Sou
nd t
rans
mis
sion
rat
io (
dB)
Integrated from 0-1kHz
Piezoceramic Actuators
Force Actuators
What Happens to the Panel Vibration?
Feedback gain
Integrated from 0-1kHz
Piezoceramic Actuators
Force ActuatorsK
ine
tic e
nerg
y (d
B)
Kin
etic
ene
rgy
(dB
)
Increasing gain
Frequency (Hz)
Experimental Result (after Bianchi et al)
• Gain limited by accelerometer resonance• Compensator used in feedback circuit
Pre
ssur
e (d
B r
e ar
bitr
ary
units
)
Concluding Remarks
• Active sound control is being used as an alternative to passive control in many different applications especially at low frequencies
• ducts
• aircraft
• automobile
• Combination of acoustic and vibration control maybe seen in the future
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