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1 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Deliberate Nonlinear Design and Control of LoudspeakersAndrew Bright
Nokia Corporation
Helsinki, Finland
Symposium “Nonlinear Compensation of Loudspeakers”
Technical University of Denmark
Dec. 2, 2004
2 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Overview
• Deliberate nonlinear design: the tradeoffs
• Controller: Architecture & general considerations
• Loudspeaker model: Alternatives & affects on controller
• Constructing a nonlinear compensation controller from a loudspeaker model
• Tuning the controller
• Optimal design with nonlinear control
3 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Typical Microspeaker
• ø15 ~ 20mm x 3~5 mm
• Standard electrodynamic motor
• Usage: Ring tones, FM radio, hands-free telephony
• Key engineering parameters• Voltage sensitivity• Size (including enclosures)
Vc
Rear volume Internal pressurep tc( )
+ ( )x td
15~20mm
2~8cc
4 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Deliberate Nonlinear Design:Shortening Coil Height
• Optimisation of coil height• Short height provides
higher sensitivity• Large height reduces
nonlinearity, sensitivity• Simulation of basic parameters
vs. coil height
this form most practical
5 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Change in basic parameters vs. coil height
VAS increases with decreasing height
Assumptions:
• Coil length proportional to height
• Constant resonance frequency
6 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Small-signal sensitivity vs. coil height
7 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Overview
• Deliberate nonlinear design: the tradeoffs
• Controller: Architecture & general considerations
• Loudspeaker model: Alternatives & affects on controller
• Constructing a nonlinear compensation controller from a loudspeaker model
• Tuning the controller
• Optimal design with nonlinear control
8 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
General considerations for a controller
• Many audio products require DSP (MP3, dig. cellular telephony)
• Analogue:+Can be designed from physical model (s-space; differential eqs.)–Drift problems–Not (easily) programmable
• Digital:+Programmable+Cheap (free?) HW – if already required– Algorithms very different from physical model
SignalProcessor
PowerAmplifier
LoudspeakerAudio input signal
processedaudio signal
poweraudio signal
acousticsound field
+-
9 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Architectures for nonlinear compensation
Feedback control
• Force y(t) to follow v(t)
Feedforward control
• Model loudspeaker, compensate accordingly
Adaptive feedforward control
• System identification to update feedforward controller
Controller
Feedforwardprocessor Plant
InputControl signal Output
v t( ) u t( ) y t( )
Controller
Plant
InputControl signal
Feedback signal
Output
v t( ) u t( ) y t( )
Plant
Plant Model
Input Control signal
Output
Plant feedback
v t( ) u t( ) y t( )
y tm( )
y tp( )Σ-
+
FeedforwardProcessor
System ID by Adaptive filtering
Controller
Plant = Loudspeaker
10 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Theoretical foundations for nonlinear compensation
Strategies (theoretical approach) for digital adaptive feedforward controller:• Volterra series, discrete-time
• Straightforward nonlinear theory: extension of linear system theory• Computationally expensive• Parameters have no physical interpretation
• System ID difficult w/o expensive feedback sensor• Narmax-model, Neural-Network
• Computationally cheaper than Volterra series• Parameters have no physical interpretation
• System ID difficult w/o expensive feedback sensor
• Feedback linearisation (inverse dynamics model)• Parameters with physical interpretation• System ID by indirect (inexpensive) feedback sensor• Same computational effort as Narmax model
Plant
Plant Model
Input Control signal
Output
Plant feedback
v t( ) u t( ) y t( )
y tm( )
y tp( )Σ-
+
FeedforwardProcessor
System ID by Adaptive filtering
Controller
Generic m
ethodsM
odel-based
methods
11 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Feedback linearisation
Digital controller design: from plant model
• Continuous time model• “Digitised” with numeric integrators• Model built from traditional LPM• Difficult parameter updating
• Discrete time model• Easy digital controller design? How to build model? (Ldspk. LPM is in continuous-time)
12 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Overview
• Deliberate nonlinear design: the tradeoffs
• Controller: Architecture & general considerations
• Loudspeaker model: Alternatives & affects on controller
• Constructing a nonlinear compensation controller from a loudspeaker model
• Tuning the controller
• Optimal design with nonlinear control
13 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
+xd
mdkd
cd
Zrm
+fc
vc
φ0
to voice coil
power amplifier
( ) ( ) ( )dt
tdxdx
txdLti
dttdx
txdt
tditxLtiRtv ddeb
cd
dc
debcebc)()(
)()(
)()(
)()()( +φ++=
Lumped-Parameter Nonlinear Loudspeaker Model
( ) ( ) ( ) ( ))(
)()()()(
)()()()()( 2
21
2
2
tdxtxdL
titxtxkdt
tdxc
dttxd
txmtxtid
debcddt
dt
ddtdc −++=φ
• Combination of electrical and mechanical LPM models
• Acoustic dynamics treated as mechanical-equivalents
• Parameter variation with displacement produces nonlinearity
• Chief culprits:• φ(x): transduction
coefficient• k(x): suspension stiffness
• Nonlinear differential equations may be analysed by numerical integration, other methods.
14 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
System dynamics view
Electrical admittance(blocked)
Mechanical receptance
(open-circuit)
Mechano-acoustic
transduction, radiation
• Linear model
)(svc )(sic )(sf c )(sp
0φebR
1
ttt kscsm
1
++2 πρ
40 sS d
+
-
0φ s
)(sx ds2
15 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Adding nonlinear components
)(svc )(sic )(sp
ttt kscsm
1
++2 πρ
40Sd
+ +
- -
φ( )x s
)(sx d
φ( )x
φ( )xebR
1 s2
x·k x1( )
)(sfc
force-factor non-uniformity suspension stiffness non-uniformity
• Added as zero-memory nonlinear systems to linear model
16 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
DSP implementation of model
• Linear filters needed for:• Mechanical receptance• Differentiation
• Other components are “zero-memory”
)(svc )(sic )(sp
πρ
40Sd
+ +
- -
φ( )x
)(sx d
φ( )x
φ( )xebR
1 s2
x·k x1( )
)(sfc
Linear filter
Linearfilter
17 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
FIR filter Mechanical Receptance
• FIR inherently stable –attractive for adaptive filtering
• Loudspeaker’s mechanical dynamics are ‘resonant’
• Inefficiently modelled by all-zero filter
Example electrical admittance impulse
response
18 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
IIR Filter for Mechanical Receptance• Resonant behaviour modelled with low-order IIR filter
• Same structure can be used to approximate differentiation• Nonlinear-elements are zero-memory systems: straightforward digital
implementation
• Forms basis for feedback linearisation-based design of nonlinear controller
Σ Σ
Σ
Σ
Σ
Σ
Σ
Σ
- -+ + f nc[ ]i nc[ ] x nd·est[ ]
u nd[ ]
v nc[ ]
z-1
z-1
z-1
bdt·0
bdt 1·
σx
adt 1·
a1
a2
1/Reb w ns[ ]φ( )x
φ( )x
k x1( )
19 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Overview
• Deliberate nonlinear design: the tradeoffs
• Controller: Architecture & general considerations
• Loudspeaker model: Alternatives & affects on controller
• Constructing a nonlinear compensation controller from a loudspeaker model
• Tuning the controller
• Optimal design with nonlinear control
20 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Inverted digital model
• Algebraic inversion of voltage & displacement described by model
• Basis for nonlinear compensation algorithm
• Describes voltage that would have created some specified displacement 1/φ
φ
( )x
( )xk x1( )
Σ Σ
Σ Σ
r np[ ]
r np[ -1]
u nd·e[ ]
r nlin[ ]
z-1
bdt·0
bdt 1·adt 1·
Σ
Σ
z-1
z-1
1/σx
a1·m
a2·m
Reb
21 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Complete nonlinear compensatorNonlinear compensator
100 1000 Fs/2-10
0
10
20
30
40
50
Pre-filtering
100 1000 Fs/2-70
-60
-50
-40
-30
-20
-10
22 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Overview
• Deliberate nonlinear design: the tradeoffs
• Controller: Architecture & general considerations
• Loudspeaker model: Alternatives & affects on controller
• Constructing a nonlinear compensation controller from a loudspeaker model
• Tuning the controller
• Optimal design with nonlinear control
23 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Parametric drift and uncertainty
• Loudspeaker characteristics change with:• Manufacturing tolerances • Temperature variations
• Parameter drift causes mistuning of feedforward controllerØ Fatal problem for pure feedforward control• Controller must be tuned to changes in the loudspeaker
Parameter name SymbolTemperature Variation
CoefficientManufacturing tolerance
DC-resistance R eb 0.004·R eb ·0°C ±10%
Suspension damping c d -0.05 ±10
Suspension stiffness k d (none) ±30%
Transduction coefficient φ0 -0.005 n/a
24 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Tracking Changes in the Loudspeaker
• System identification: Tracking changes using adaptive filtering
Poweramplifier
u t( )
i tc( )
u td( )
p t( )Σ
ε[ ]n
shunt res.
ic
vc
vc
Adaptive filter(plant model)
Loudspeaker
Plant
-
• Identified parameters are updated in pre-processor
• Measurement of electrical current, mechanical vibration, or acoustic pressure can be measured
25 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Overview
• Deliberate nonlinear design: the tradeoffs
• Controller: Architecture & general considerations
• Loudspeaker model: Alternatives & affects on controller
• Constructing a nonlinear compensation controller from a loudspeaker model
• Tuning the controller
• Optimal design with nonlinear control
26 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Simulation of effective sensitivity
)(Pk)(Pk
wu
Cc =
• Effective sensitivity calculated as a function of coil height calculated
• Cc is additional amplifier output required for distortion compensation
0~
1
)()(1
xc
m
ceff sv
spC
S =
27 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
486Hz
28 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
825Hz
29 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
1074Hz
30 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
3082Hz
31 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Voice Coil Height
1.2 mm
0.8 mm
0.4 mm
0.2 mm
Original Coil Height
Coil Height = 2 × Magnet Gap
Coil Height = Magnet Gap
Coil Height = ½ × Magnet Gap
Plastic insert
Plastic insert
Plastic insert
Measurement
• Modified voice-coil height samples prepared for measurement
• 1.2mm (standard height)• 0.8mm• 0.4mm• 0.2mm
32 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Linear FRF of shortened-height V-C samps.
• Shorter heights provide higher sensitivity
• 0.2mm shows ~10dB higher voltage sensitivity, +4dB power efficiency
33 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Compensation of distortion
Measurements from 0.2mm height coil• Broken: No control• Sold: With control
34 © NOKIA Presentation_Name.PPT / DD-MM-YYYY / Initials Company Confidential
Conclusions
• Deliberate introduction of nonlinearity can increase loudspeakersensitivity
• Nonlinearities can be compensated by digital processing
• Optimal design point not fully clear
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