The Power of Loudspeaker Models -...

The power of Loudspeaker ModelsThe power of Loudspeaker Models, 117th AES Convention, 1

The power of Loudspeaker ModelsThe power of Loudspeaker Models

CHAIRMAN: W. Klippel

PANEL: Richard Small, David Clark, Jürgen Ringlstetter, Andrew Bright

AES 117th CONVENTION, OCTOBER 28-31, SAN FRANCISCO


Road Map of the WorkshopRoad Map of the Workshop

1) Basics of loudspeaker modeling

2) Modeling at small and large amplitudes

3) Measurement of model parameters

4) Applications (analysis, synthesis, control)

5) Phenomena not modelled so far



Basics of Basics of Loudspeaker ModelingLoudspeaker Modeling

InputSignal

Output Signal

MODEL

Real system

image of image of realityreality


Modeling Modeling –– Abstraction of the realityAbstraction of the reality

dtixLd

dtdxxBliRu e

e))(()( ++=

xxKdtdxR

dtxdMii

dxxdLxBl msmsms

e )()()( 2

2

++=

+

Differential Equation

• data reduction

• no complete description

• preserving only relevant features

H(jω)Transfer function

magnetic flux

FEMBEM

MMSCMS(x) RMS

b(x)

LE(x)RE(TV)

v

Fm(x,i)

i

b(x)v b(x)i

L2(x)

R2(x)u

Lumped Parameter Model

Abstraction


Definition of Terms Definition of Terms

InputSignal

Output SignalMODEL

The model is characterized by

1. Structure (equivalent circuit)

2. Free Parameters (Cms, Bl, Mms, ...)

3. State variables (displacement x, ...)


1. 1. Structure Structure of of the the Model Model

InputSignal

Output SignalMODEL

• gives general description of the physical mechanisms

• depends on the scope (micro or macroscopic view)

• is restricted to the transducer principle

MMSCMS(x) RMS

b(x)

LE(x)RE(TV)

v

Fm(x,i)

i

b(x)v b(x)i

L2(x)

R2(x)u

Lumped Parameter Model dtixLd

dtdxxBliRu e

e))(()( ++=

xxKdtdxR

dtxdMii

dxxdLxBl msmsms

e )()()( 2

2

++=

+



2. Model Parameters 2. Model Parameters Input

SignalOutput SignalMODEL

Parameters:

• describe the properties of the particular unit

• are constant values or functions of one or more variables

• should be independent of input and measurement conditions

H(jω)

MmsSd Kms(x)Bl(x)h2(t1,t2)

h3(t1,t2 ,t3)geometryMaterial

parameters


3. State Variables3. State Variables

InputSignal

Output SignalMODEL

• describe the instantaneous state

• vary with time

• depend on input signal

Current i(t)

Sound pressure p(t)

Temperature T(t)

Displacement x(t)


Can we consider harmonic distortion as parameters Can we consider harmonic distortion as parameters ??

InputSignal

Output Signal

MODEL

SpectralAnalysis Symptoms

• Harmonic distortion (HD2, THD)• Intermodulation (IMD)• Compression of Fundamental

• DC displacement

No, they depend on the stimulus used !but they might be meaningful characteristics of the loudspeaker


How can we assess the How can we assess the limits of a limits of a model model ??

PredictedOutput

MODEL

Measured Output

InputSignal

Agreement ?Agreement ?

Ambient condition









WhatWhat kindskinds of of models models do do we needwe need ??

voice-coil

displacement

30

3

10

1

0,3

X[mm]

Amplitude DRIVER MODELING

weakly nonlinear

strongly nonlineartime-variant

Small signal domain

Destruction

Linear Model

Nonlinear Model


Subjective SensationOutput Spectrum

Linear Linear ModelModel

Spectral discoloration

Impulse accuracy

FundamentalComponents

Input Spectrum

Fundamentalschanged in amplitude and phase

LinearSystemInput

SignalOutputSignal


Linear Linear lumped parameter modelinglumped parameter modeling

Olson 1950


Progress in Linear Progress in Linear Modeling Modeling

Important steps:• electrical analogies Olson, Beranek, ...• optimal system design Thiele, Small, ...• visco-elastic behavior Knudsen, ...• voice coil impedance Wright, Leach, ...


Models Models forfor Electrical Impedance Electrical Impedance ZLZL

• Leach

ZL(jω)= Krm·ωErm + j·(Kxm·ωExm )• Wright

MmsCms Rms

Bl

Re

vI

Blv BliU

ZL(jω)

• LR-3 (shunted inductance) Le

R2

c)

L2

R3

L3

ZL(jω)= K·(jω)n ; ω= 2πf

a)ZL(jω,x)

ZL(jω) = Le·jω + (R2·L2·jω ) / (R2 + L2·jω)• LR-2 (shunted inductance

Le

L2

R2

b)


DEMODEMO

• Fitting the electrical impedances over a wide frequency range


DiscussionDiscussion::

How much accuracy do we need ?How much accuracy do we need ?


KLIPPEL

0

5

10

15

20

25

101 102 103 104

Magnitude of electric impedance Z(f)[O

hm]

Frequency [Hz]

x= 0 mm x = - 4 mm x = + 4 mm

Variation of Variation of electrical Impedance with Displacementelectrical Impedance with Displacement

X=X=--4 mm4 mm

X=4 mmX=4 mm


Linear ModelsLinear ModelsBenefits:

• Simple to use and to understand

• Minimal number of parameters

• Easy to solve by numerical methods

Drawbacks:

• Limited to the small signal domain

• Can not explain nonlinear effects (distortion, compression, instabilities)


Subjective SensationOutput Spectrum

NonlNonlinear inear ModelModel

Spectral discoloration

Impulse accuracy

FundamentalComponents

Input Spectrum

Fundamentalschanged in amplitude and phase

NonlinearSystemInput

SignalOutputSignal

New spectral Components

Disturbances

Amplitude compression


Effect Effect of of the Nonlinear the Nonlinear SuspensionSuspensionf<<fs

Olson 1950


x = Cms(x) F

x

0

Compliance CCompliance Cmsms(x)(x)

KLIPPEL

0,00

0,25

0,50

0,75

1,00

1,25

Cms [mm/N]

--10,0 -7,5 -5,0 -2,5 0,0 2,5 5,0 7,5 10,0 << Coil in X [mm] coil out >>

Cms(x) determined by

• suspension geometry

• impragnation

• adjustment of spider and surround


Signal Signal flow chart describing effect flow chart describing effect of of KmsKms(x)(x)

fsVoltage pressure

highpass

fslowpass

Displacement x

multiplier

distortion

KmsKms(x)(x)--KmsKms(0)(0)

Multiplication of signals nonlinear distortion


Force Factor Bl(x)Force Factor Bl(x)

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

5,5

-7,5 -5,0 -2,5 0,0 2,5 5,0 7,5

N

/

A

x [mm]

force factor

permanent flux Φ0

Voice coil

magnet

Pole piece

Bl(x) determined by

• Magnetic field distribution

• Height and overhang of the coil

• Optimal voice coil position


Voice Coil Inductance LVoice Coil Inductance Lee(x)(x)

voice coil 0,00

0,25

0,50

0,75

1,00

-7,5 -5,0 -2,5 0,0 2,5 5,0 7,5

mH

x [mm]

inductance

LLee(x) determined by(x) determined by

• geometry of coil, gap, magnet

• optimal size and position of short cut ring


DiscussionDiscussion

Which nonlinearities areWhich nonlinearities are relevant and relevant and should be modelled should be modelled ??


Criteria for dominant Criteria for dominant NonlinearitiesNonlinearities

• limits acoustical output• generates audible distortion• indicates an overload situation• related with cost, weight, volume,

efficiency









Parameter Measurement

dtixLd

dtdxxBliRu e

e))(()( ++=

xxKdtdxR

dtxdMii

dxxdLxBl msmsms

e )()()( 2

2

++=

+


Identification

Particular unit

Kms(x)MmsSd

Bl(x),

Valid for the particular unit


Method:

• Apply small ac-stimulus

• Measure state variables • (voltage, current, displacement)

• Estimate optimal parameters (by fitting the linear model)

• Perturbate loudspeaker to dispense with mechanical sensor

Dynamical Measurementof Small Signal Parameters

F

x

tangent

KMS(x=0)=FACxAC


SmallSmall--Signal MeasurementsSignal Measurements• What mounting conditions?

– What is “free air”? – Is the mounting rigid?– Is a baffle better?

• Axis Horizontal or vertical?• What signal level?

– How small?– Is it linear?

• Voltage or current drive?• Add mass, stiffness or laser?


Measure The Moving SystemMeasure The Moving SystemDon’t include the motor and chassis!


The Chassis Must Not MoveThe Chassis Must Not MoveOK with or without baffle if rigid


Axis Horizontal or Vertical?Axis Horizontal or Vertical?

Xg = 25 / fS2 cm (Olson)


What is a Small Signal?What is a Small Signal?

• Some standards and specifications require “one watt” – often not linear

• We know fS can vary with level(CMS is not a constant)

• Do not expect same result from different methods (different levels)

• Box design is still possible!


Voltage vs. Current DriveVoltage vs. Current Drive

• “No difference” IF “linear”• Current drive allows greater excursion

around resonance• But current-drive methods usually very

low level (good chance of linearity)• Calculation method must take

measurement method into account


That Extra Measurement (1)That Extra Measurement (1)• Added Stiffness:

account for altered mass

VAS/VB = (fC2/fS2) – 1 (no mass change)

VAS/VB = (fCQEC/fSQES) – 1 (Thiele)

(May be hard to assess added volume accurately)


That Extra Measurement (2)That Extra Measurement (2)• Added Mass:

account for altered compliance

MMS/MAM = {(fS2/fAM2) – 1}-1 (no compliance change)

MMS/MAM = {(fSQEAM/fAMQES) – 1}-1 (corrected for C shift)

(Only valid if Bl does not change!)


That Extra Measurement (3)That Extra Measurement (3)

• Added Luxury: with displacement laser,– no disturbance to the driver– all data can be collected at same time



Is it useful to apply a linear model to a loudspeaker operated in the large signal domain and to work with "effective" parameters ?


UsingUsing a linear a linear model model in in the the large signal domainlarge signal domain

PROBLEM: „Effective Parameters“ depend on stimulus !!

KLIPPEL

0.00

0.25

0.50

0.75

1.00

1.25

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

stiffness K_MS(x)

K_M

S [N

/mm

]

<< coil in x [mm] coil out >>

-x_max < x < x_max

KLIPPEL

0.0

0.5

1.0

1.5

2.0

2.5

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

force factor b(x)

b [N

/A]


-x_max < x < x_max

KLIPPEL

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

inductance L_E(x)

L_E

[mH

]


-x_max < x < x_max

Bl ≠const. Le(xpeak) ≠const. Kms≠const.


Method:

• Sample the working range

• Generate DC displacement xDC

• Measure associated state signals FDC

• Calculate instantaneous parameters

• Repeat the measurement at other working points

Static Large Signal Measurement

F

x

secant

KMS(x)= FDCxDC


Static measurement techniqueStatic measurement technique


Quasi-static Measurement

Method:

• Sample the working range • Generate a variable offset XDC

• Excite with small AC-signal• Measure state variables XAC and FAC

• Calculate gradient Kgrad (XDC)• Repeat measurement at other

working points• Parameter Transformation

Kgrad(x)=FACXAC

Transformation

KMS(x)=FX

xAC

tangent

xDC

FDC FAC


Measuring Loudspeaker Measuring Loudspeaker ExcursionExcursionDavid ClarkDLC Design


Music Excursion PictureMusic Excursion Picture


DriveDrive--Unit Measurement at Unit Measurement at Excursion (DUMAX)Excursion (DUMAX)

• Moves cone to measurement position by air pressure

• Measures Bl as a function of X• Measures Cms as a function of X• Measures Z as a function of X• Measures other parameters

– Moving mass– Mechanical damping– DCR– Diaphragm area


Picture of DUMAXPicture of DUMAX

Pressure source

Electronics unit

Computer

Pressure chamber

Small adaptor boards

DUT

Laser

I/O connections


DUMAX specDUMAX specMfg., Model Sonavox XW7F-18808-AB 0..18 6# Outside Dimensions

Max Diameter (mm): 204Description: 5x7" Full Range Min Diameter (mm): 145

Mounting Depth (mm): 56Sample # Sample #4 Magnet Diameter (mm): 73

Driver weight (Kg): 0.78Report date: 03/14/03 Technician: Ponte

Thiele-Small Parameters @ x=0 Electro-Mechanical Parameters @ x=0Fs 74.97 Hz Mmd 6.76 g By Fs OscillationQe 1.13 Cms 0.0006 m/N 0.570874 mm/NQm 3.94 Rms 0.94 ohmmQts 0.88 Sd 0.0159 m^2 effective radiusVas 20.27 ltr. Bl 3.35 N/A #VALUE! mmSd 0.0159 m^2 Re 3.42 ohmRe 3.42 ohm

X Parameters from Curve FitXmag 2.7 mm Xsus 5.1 mm Xmax 2.7Mag Maximum -0.2 mm Sus Minimum -0.2 mm

Bl fit between Kms fit between-3 mm and 2.5 mm -5 mm and 5.5 mm

Bl vs. X

0

0.5

1

1.5

2

2.5

3

3.5

4

-12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5displacement (mm)

Bl (

N/A

)

BL vs. x (measured)71% of rest BL value (Xmag)Bl quadratic curve fit

Incremental Kms vs. X

0

1

2

3

4

5

6

7

8

9

10

-12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5displacement (mm)

Km

s (N

/mm

)

Kms vs x (measured)400% (4X) of rest Kms ( Xsus)Kms 4th order curve fit


DUMAX specsDUMAX specsMfg., Model Panasonic, A00-6-036H Outside Dimensions

Max Diameter (mm): 202Description: 6.5" Poly Dual Cone Min Diameter (mm): 158

Mounting Depth (mm): 36Sample # 28156 7Z800 Magnet Diameter (mm): 65.5

Driver weight (Kg): 0.59Report date: 10/21/03 Technician: Busch

Thiele-Small Parameters @ x=0 Electro-Mechanical Parameters @ x=0Fs 63.13 Hz Mmd 7.06 g By Fs OscillationQe 0.99 Cms 0.0008 m/N 0.827591 mm/NQm 3.54 Rms 0.86 ohmmQts 0.77 Sd 0.0106 m^2 effective radiusVas 13.13 ltr. Bl 2.39 N/A 58.17 mmSd 0.0106 m^2 Re 1.85 ohmRe 1.85 ohm

X Parameters from Curve FitXmag 2.7 mm Xsus 3.3 mm Xmax 2.7Mag Maximum 0.2 mm Sus Minimum 0.3 mm

Bl fit between Kms fit between-3 mm and 3 mm -5 mm and 5 mm

Bl vs. X

0

0.5

1

1.5

2

2.5

3

-12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5displacement (mm)

Bl (

N/A

)

BL vs. x (measured)71% of rest BL value (Xmag)Bl Eff curve fit


0

2

4

6

8

10

12

-12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5displacement (mm)

Km

s (N

/mm

)


Mfg., Model Panasonic, A00-6-036H Outside Dimensions Max Diameter (mm): 202

Description: 6.5" Poly Dual Cone Min Diameter (mm): 158 Mounting Depth (mm): 36

Sample # 28156 7Z800 Magnet Diameter (mm): 65.5 Driver weight (Kg): 0.59

Report date: 10/21/03 Technician: Busch

Thiele-Small Parameters @ x=0 Electro-Mechanical Parameters @ x=0Fs 63.13 Hz Mmd 7.06 g By Fs OscillationQe 0.99 Cms 0.0008 m/N 0.827591 mm/NQm 3.54 Rms 0.86 ohmmQts 0.77 Sd 0.0106 m^2 effective radiusVas 13.13 ltr. Bl 2.39 N/A 58.17 mmSd 0.0106 m^2 Re 1.85 ohmRe 1.85 ohm

X Parameters from Curve FitXmag 2.7 mm Xsus 3.3 mm Xmax 2.7Mag Maximum 0.2 mm Sus Minimum 0.3 mm

Bl fit between Kms fit between

Bl vs. X

0

0.5

1

1.5

2

2.5

3

-12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5displacement (mm)

Bl (

N/A

)

BL vs. x (measured)71% of rest BL value (Xmag)Bl Eff curve fit


0

2

4

6

8

10

12

-12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5displacement (mm)

Km

s (N

/mm

)



DUMAX SIMULATED DUMAX SIMULATED EXCURSIONEXCURSIONBegin


+10 mm+10 mm


+5 mm+5 mm


0 mm0 mm


--5 mm5 mm


--10 mm10 mm


Interpreting Excursion DataInterpreting Excursion Data

• Bl falloff from rest position translates directly to AMD– Non-linear suspension reduces AMD– Stiffer suspension reduces AMD

• Sharp transition in Bl or Cms produces Hi-2– Gradual non-linearities best

• Cms width less than Bl:– Non-utilization of motor capability– Possible speaker damage


Method:

• Excite speaker with large AC-signal

• Measure state variables• Estimate parameters to describe the

relationship between state variables

Dynamic Large Signal MeasurementF

x

secant

KMS(x)=FX

FAC(t)

XAC(t)


Loudspaaker Loudspaaker systemsystem

Distortion Analyzer

amplifier

Large Large Signal Signal IdentificationIdentification

• gives electrical, mechanical, acoustical parameters• for any electrodynamical transducer• mounted in Closed or vented enclosures, horns, ...• monitors voltage + current only • long term measurement • music as stimulus• real time distortion analysis



Are systematic differences in the results of static, quasi-static or dynamic measurements ?









ApplicationsApplications

Targets:

• Equalization of the linear amplitude and phase response

• Linearization (compensation of distortion)

• Mechanical and thermal Protection

• On-line Diagnosis

1. Electrical Control of Loudspeaker System

Audio signal

controller

Modelparameters


Electrical Control of LoudspeakersElectrical Control of Loudspeakers

Enable, in real-time:• Equalisation• Protection• Nonlinear compensation• Tuning & diagnostic

Power amplifier

Nonlinearcompensation

Systemmeasurement

ProtectionEQTransducerLevel & Trans.

DRC

Transducer &acoustics

DSP

Tuning, diagnostic


ModelModel--based algorithmsbased algorithmsGeneric algorithms• NARMAX, Volterra series, Neural-network

algorithms– Long identification times (1 ~ 24 hours)– Parameters, states have no physical interpretation

• Do not predict physical displacement, temperature, etc.

Model based algorithms• Pole-zero filters + zero-memory nonlinear

systems– Familiar parameters– Predict “physical states,” e.g. displacement,

temperature, etc.


Transducer ProtectionTransducer Protection

• Protection based on real-time ‘state’ prediction (temperature, displacement)

• Enables re-specification (de-rating) of loudspeaker input limits

X-Prot/DispX-Prot/Therm

Temperaturepredictor

Temperaturelimitter

Input OutputDisplacement

limitter

Displacementpredictor


Tuning & DiagnosticTuning & Diagnostic

• Adaptive filter performs system identification, to track changes in model parameters of a model of the loudspeaker

• Identified model parameters used by protection & compensation algorithms

Poweramplifier

u t( )

i tc( )u td( )p t( )

Σ

ε[ ]n

shunt res.

ic

vc

vc

Adaptive filter(plant model)

Loudspeaker

Plant

-


Transducer Transducer nonlinearitynonlinearity compensationcompensation

• Based on inverted loudspeaker model• Frees motor design for sensitivity optimisation

1/φ

1/φ

( )x

( )xk x1( )

Σ Σ

Σ Σ

x nd[ ]

x nd[ -1]

u nd[ ]

v nc[ ]

z-1

bdt·0

bdt 1·adt 1·

Σ

Σ

z-1

z-1

1/σx

a1

a2

Linear dynamics Reb


2. Parameter Measurement

Targets

• Specification of loudspeaker systems

• Interface between driver and system design

• Quality Control

dtixLd

dtdxxBliRu e

e))(()( ++=

xxKdtdxR

dtxdMii

dxxdLxBl msmsms

e )()()( 2

2

++=

+


Identification

Particular unit Kms(x)MmsSd

Bl(x),



symbol name category typ value unit

f c loudspeaker cut-off freq. 950 HzQc loudspeaker Q-value 8.5mt total moving mass 6.00E-05 kgR eb DC-resistance 7.2 Ohmφ0 transduction coefficient (B·l) 0.563 Tm, Wb/m, N/Ax lm m displacement limit 0.0004 m

R tv voice-coil thermal res. 100 °C/WCtv voice-coil thermal cap. 0.006667 J/°CR tm magnet thermal res. 25 °C/WCtm magnet thermal cap. 0.5 J/°CT lm m max. voice-coil temp. 100 °Cφ1 phi - 1 150φ2 phi - 2 -1.00E+06φ3 phi - 3φ4 phi - 4k 1 k - 1k 2 k - 2k 3 k - 3k 4 k - 4

loudspk. & cab.

loudspk.

Supplier A

Integrator

Supplier B

Supplier C

Transducer A 1

Product 1

Transducer B 1

Transducer C 1

Transducer A 2

Product 2

Transducer B 2

Transducer C 2

Transducer A N

Product N

Transducer B N

Transducer C N

...

...

......

Information management

InterInter--company communicationcompany communication• Model provides a structure for inter-

company information management• Thiele-Small Parameters form a basis• Simple extensions possible for

– Nonlinear characteristics– Thermal behaviour



Targets

• Prediction of the Transfer Behavior

• Relationship between causes and symptoms

• Investigation of design choices

3. Synthesis and Diagnosis of Loudspeakers

MODELStimulus(Music)

Kms(x)MmsSd

Bl(x),Parameters

Symptoms(Distortion)Output


Parameters Parameters are are a a „„common languagecommon language““between between different different toolstools

Linear DesignBox, Cone, Crossover

Parametermeasurement

KLIPPEL

0,0

0,5

1,0

1,5

2,0 2,5

3,0

3,5 4,0

4,5

5,0

5,5

-10,0 -7,5 -5,0 -2,5 0,0 2,5 5,0 7,5 10,0

force factor b(x)

b [N/A]


-x_max < x < x_max

FEM motor design

electrical

Power Test

KLIPPEL

0

25 50

75

100

125

0,5

1,0 1,5 2,0

2,5

3,0

3,5

4,0

100 200 300 400 500 600 700

Delta Tv [K]

P [W]

t [sec]

Delta Tv P

thermal

mechanical

Vibration (FEM)Radiation (BEM)

Acoustical Measurement

-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0 1 2 3 4 5 6 7 8 9 10

Stimulus (t) vs time

[V]

Time [ms]

Stimulus (t)

Room Simulation

acousticalParameters

SymptomsLarge signal behavior

SIM



4. Auralization

Targets:

• Objective Assessment with music

• Subjective Evaluation of sound quality

• Optimization of cost/performance ratio

• Tuning to the market


AuralizationAuralization in in Loudspeaker DevelopmentLoudspeaker Development

Subjective Evaluation• Personal Impression• Sufficient Sound Quality• Tuning to the target market • Performance/Cost Ratio

Objective Evaluation• Distortion, Maximal Output • Displacement, Temperature• Evaluation of Design Choices• Indications for Improvements

MarketingManagement

DevelopmentManufacturing


Participate Participate in in Interactive Interactive Listening Listening TestTest

AURA

report

ServerMP3files

judmements

Test signalMusic

statesdistortion

Parameters results

High quality

Internet

wwwwww.klippel.de.klippel.de









Some unmodelled PhenomenaSome unmodelled Phenomena

1. Other visco-elastic effects• K(x=0, xpeak)

2. „flux“ modulation• Inductance L(x,i)• Force factor Bl(x,i)


Dependency of Cms(f) on FrequencyDependency of Cms(f) on Frequency

f [Hz]

10 20 50 100 200 500 1k 2k 5k 10k

Magnitude of transfer function Hx(f)= X(f)/U(f)

[mm

/V]

Frequency [Hz]

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Measured Fitted without creep Creep model


Considering CreepConsidering Creep

• Dissipative model

• Non-dissipative model

−=

smsms f

fCfC 10log1)( λ

MmsC1 Rms

Bl

Re

vI

Blv BliU

ZL(jω)

C2

R2

Kundsen and Jensen, JAES 1993

MmsCms(f) Rms

Bl

Re

vI

Blv BliU

ZL(jω)


Interaction Interaction betweenbetween NonlinearitiesNonlinearities

Bl(x)

Le(x)

Cms(x)

Creepe.g. λ-parameter

DC Force

The dc displacement may be increased by creep

Cms(f=0) DC Displacement


Dependency of resonance on peak displacementDependency of resonance on peak displacement

27 % variationIn small signal domain

0,1 1 1040

50

60

70

80

90

100

110

fs

Peak value of displacement mm

(linear model)

Hz

Small Signal Domain55 Hz

71 Hz

Large Signal Domain

resonance resonance frequencyfrequency


Dependency on Peak DisplacementDependency on Peak Displacement

significant variation at the rest position x=0

0,0 0,5 1,0 1,5 2,0 2,5 3,0

-20 -15 -10 -5 0 5 10 15 20

[N/mm]

Displacement x [mm]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

-20 -15 -10 -5 0 5 10 15 20

[N/mm]

Displacement x [mm]

voltage

0,0 0,5 1,0 1,5 2,0 2,5 3,0

-20 -15 -10 -5 0 5 10 15 20

[N/mm]

Displacement x [mm]

voltage

0,0 0,5 1,0 1,5 2,0 2,5 3,0

-20 -15 -10 -5 0 5 10 15 20

[N/mm]

Displacement x [mm]

voltage


Mechanical properties of loudspeaker suspension parts

this points to visco-elastic behaviour

How can we describe visco-elasticity?

Relation between force and deflection not one-to-one

Here: two separately measured curves(for positive and negative direction)

Hysteresis: relation between force and deflectiondependent on previous history


1) Creep experiment:

A certain stress σ0 is applied for a certain timeand the strain ε is measured

Elastic: strain follows stress (Hooke)

ε = J * σ0 (J = compliance)

Visco-elastic: one part of the strain occursinstantaneously, another part whilethe stress is applied (creeping)

σ

t

t

ε

t

ε

σ0


1) Creep experiment:

Delivers creep compliance J(t)

ε = σ0* J0 t = t0

J0 = elastic compliance

ε(t) = σ0 * J(t) t0 < t < t1

J(t) = creep compliance t

ε

t0 t1

t

σ0

σ


2) Relaxation experiment:

A certain strain ε0 is appliedand the stress σ is measured

Elastic: stress follows strain (Hooke)σ = E * ε (E = Young´s modulus)

Visco-elastic: stress reaches an initial value anddecreases subsequently (relaxation)

t

σ

σ

t

t

ε

ε0


2) Relaxation experiment:

Delivers relaxation modulus E(t)

σ = σ0 t = t0

σ (t) = ε0 * E(t) t0 < t < t1

E(t) = relaxation modulus

t

ε

ε0

t

σ

t0

σ0

t1


Can we observe these visco-elastic phenomena in loudspeaker suspension parts?

Two visco-elastic phenomena:

1) Creep

2) Relaxationt

ε

t0 t1

t

σ

t0 t1


Creep measurement of a 80mm rubber surround:


Relaxation measurement of a 80mm rubber surround:


Green: 12mm/s

Red: 2mm/s

Blue: 0.2mm/s

Influence of driving speed on hysteresis curve


The effectsThe effects ofof viscoelastic viscoelastic behaviorbehavior inin suspension suspension

components components

Effects in the small signal domain:• Dependency of stiffness on frequency • Dependency of observed resonance on peak displacement • Creep and relaxation (characteristic slow step response)

Effects in the large signal domain:

• The dc displacement is increased by creep

• K(x) nonlinearity is increased by creep (decreasing K(x=0)


Two points of view:

1) User:

distortion audible?

2) Engineer:

performance correlates with prediction?

visco elasticity limits validity of static measurement for

comparison with target curves from FEM simulation


What is What is „„flux modulationflux modulation“ ?“ ?

• Interaction between permanent field generated by magnet and alternative field generated by current

• Parameters depend not only on displacement but also on current

• Two mechanisms: Le(x,i) and Bl(x,i)


Voice Coil Inductance LVoice Coil Inductance Lee(i)(i)

voice coil

LLee(i) (i) depends on

• Material (permeability) and geometry of iron path

• voice coil height, number of windings

B

H

i=0

i= -10 Ai=10 A

Nonlinear relationship


Flux Modulation by Bl(i,x)Flux Modulation by Bl(i,x)

Voice coil

magnet

Pole piece

Linear Superposition of ΦA(i) and Φ0

Current i

2,5

3,0

3,5

4,0

4,5

5,0

-10,0 -7,5 -5,0 -2,5 0,0 2,5 5,0 7,5 10,0

Force factor Bl(x,i)

Bl [

N/A

]

Displacement X [mm]

i=0 i= - 10 A i = 10 A

permanent flux Φ0

alternating flux ΦA(i)


Flux modulation pictureFlux modulation picture

-3.5 -3

-2.5 -2

-1.5 -1

-0.5 0

0.5 1

1.5 2

2.5 3

3.5 -6

-2

2

6 10

0

2

4

6

8

10

Bl (N/amp)

Current (amps.)

displacement (mm)

Bl vs Current and Displacement


SummarySummary

• Models are not static but evolving• Models give a deeper understanding of

the physics• Models may be created in different

forms• Models are essential for measurement

prediction, auralization and control


Do Do youyou knowknow ALMA ?ALMA ?

Get Information on the

International Loudspeaker Manufacturer Association (ALMA)

Where: ThirstyBear Restaurant, (611 Howard Street)

When: Today, starting 4:30

The Power of Loudspeaker Models -...

Documents

Transcript of The Power of Loudspeaker Models -...