active control of sound professor mike brennan institute of sound and vibration research university...

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


-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 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


( )

2o o

p

p x

cq

S

x

9 16d 5 8d

3 4d 15 16d


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


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


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


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

qq

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


controller

Secondary source

Errorsensor


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


controller

Secondary source

Errorsensor


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)


www.bombardier.com

Periodic excitation generates intense harmonic soundfield inside cabin


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


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


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.



Frequency (Hz)

dB Active control off

Active control on



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


Force Actuators

What Happens to the Panel Vibration?

Feedback gain

Integrated from 0-1kHz


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

active control of sound professor mike brennan institute of sound and vibration research university...

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