Road Map of the Workshop

Transcription

1 The power of Loudspeaker Models PANEL: CHAIRMAN: W. Klippel Richard Small, David Clark, Jürgen Ringlstetter, Andrew Bright AES 117th CONVENTION, OCTOBER 28-31, SAN FRANCISCO The power of Loudspeaker Models, 117th AES Convention, 1 Road 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 The power of Loudspeaker Models, 117th AES Convention, 2

2 i Basics of Loudspeaker Modeling Input Signal Output Signal Real system MODEL image of reality The power of Loudspeaker Models, 117th AES Convention, 3 Modeling Abstraction of the reality Abstraction L 2 (x) magnetic flux FEM BEM R E (T V ) L E (x) C MS (x) M MS R MS F m (x,i) v R 2 (x) u b(x)v b(x) b(x)i 2 dle ( x) d x dx Bl( x) + i i = Mms + Rms + Kms( x) x 2 dx dt dt dx d( Le ( x) i) u = Rei + Bl( x) + Lumped Parameter Model dt dt Differential Equation H(jω) Transfer function data reduction no complete description preserving only relevant features The power of Loudspeaker Models, 117th AES Convention, 4

3 i R E (T V ) u L E (x) L 2 (x) R 2 (x) b(x)v b(x) v b(x)i C MS (x) F m (x,i) Definition of Terms Input Signal MODEL Output Signal The model is characterized by 1. Structure (equivalent circuit) 2. Free Parameters (C ms, Bl, M ms,...) 3. State variables (displacement x,...) The power of Loudspeaker Models, 117th AES Convention, 5 1. Structure of the Model Input Signal MODEL Output Signal M MS R MS 2 dle ( x) d x dx Bl( x) + i i = Mms + Rms + Kms( x) x 2 dx dt dt Lumped Parameter Model dx d( Le ( x) i) u = Rei + Bl( x) + dt dt Differential Equation gives general description of the physical mechanisms depends on the scope (micro or macroscopic view) is restricted to the transducer principle The power of Loudspeaker Models, 117th AES Convention, 6

4 2. Model Parameters Input Signal MODEL Mms Sd Kms(x) h 2 (t 1,t 2 ) Bl(x) H(jω) h 3 (t 1,t 2,t 3 ) Material geometry parameters Output Signal 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 The power of Loudspeaker Models, 117th AES Convention, 7 3. State Variables Input Signal MODEL Output Signal Displacement x(t) Sound pressure p(t) Temperature T(t) Current i(t) describe the instantaneous state vary with time depend on input signal The power of Loudspeaker Models, 117th AES Convention, 8

5 Can we consider harmonic distortion as parameters? Input Signal Output Signal Spectral Analysis Symptoms MODEL 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 The power of Loudspeaker Models, 117th AES Convention, 9 How can we assess the limits of a model? Input Signal MODEL Ambient condition Predicted Output Agreement? Measured Output The power of Loudspeaker Models, 117th AES Convention, 10

6 Road 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 The power of Loudspeaker Models, 117th AES Convention, 11 What kinds of models do we need? Amplitude DRIVER MODELING X [mm] ,3 Destruction strongly nonlinear time-variant weakly nonlinear Small signal domain Nonlinear Model Linear Model voice-coil displacement The power of Loudspeaker Models, 117th AES Convention, 12

7 Linear Model Input Signal Linear System Output Signal Input Spectrum Fundamental Components Output Spectrum Fundamentals changed in amplitude and phase Subjective Sensation Spectral discoloration Impulse accuracy The power of Loudspeaker Models, 117th AES Convention, 13 Linear lumped parameter modeling Olson 1950 The power of Loudspeaker Models, 117th AES Convention, 14

8 Progress in Linear Modeling Important steps: electrical analogies Olson, Beranek,... optimal system design Thiele, Small,... visco-elastic behavior Knudsen,... voice coil impedance Wright, Leach,... The power of Loudspeaker Models, 117th AES Convention, 15 Models for Electrical Impedance ZL R e C ms M ms R ms Leach Z L (jω)= K (jω) n ; ω= 2πf I U Z L (jω) Blv Bl v Bli LR-2 (shunted inductance Z L (jω) = L e jω + (R 2 L 2 jω ) / (R 2 + L 2 jω) Z L (jω,x) L e L 2 a) Wright Z L (jω)= K rm ω Erm + j (K xm ω Exm ) R 2 b) L 2 L 3 LR-3 (shunted inductance) L e R2 c) R 3 The power of Loudspeaker Models, 117th AES Convention, 16

9 DEMO Fitting the electrical impedances over a wide frequency range The power of Loudspeaker Models, 117th AES Convention, 17 Discussion: How much accuracy do we need? The power of Loudspeaker Models, 117th AES Convention, 18

10 Variation of electrical Impedance with Displacement Magnitude of electric impedance Z(f) 25 x= 0 mm x = - 4 mm x = + 4 mm KLIPPEL 20 [Ohm] X=-4 4 mm 5 X=4 mm Frequency [Hz] The power of Loudspeaker Models, 117th AES Convention, 19 Benefits: Linear Models 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) The power of Loudspeaker Models, 117th AES Convention, 20

11 Nonlinear Model Input Signal Nonlinear System Output Signal Input Spectrum Fundamental Components Output Spectrum Fundamentals changed in amplitude and phase New spectral Components Subjective Sensation Spectral discoloration Impulse accuracy Amplitude compression Disturbances The power of Loudspeaker Models, 117th AES Convention, 21 Effect of the Nonlinear Suspension f<<fs The power of Loudspeaker Models, 117th AES Convention, 22 Olson 1950

12 Compliance C ms (x) 0 x 1,25 Cms 1,00 [mm/n] 0,75 0,50 KLIPPEL x = Cms(x) F 0,25 0, ,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 The power of Loudspeaker Models, 117th AES Convention, 23 Signal flow chart describing effect of Kms(x) Voltage distortion fs highpass pressure Displacement x fs lowpass multiplier Kms(x) (x)-kms(0) Multiplication of signals nonlinear distortion The power of Loudspeaker Models, 117th AES Convention, 24

13 Force Factor Bl(x) magnet Pole piece permanent flux Φ 0 Voice coil A / 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0 N force factor -7,5-5,0-2,5 0,0 2,5 5,0 7,5 x[mm] Bl(x) determined by Magnetic field distribution Height and overhang of the coil Optimal voice coil position The power of Loudspeaker Models, 117th AES Convention, 25 Voice Coil Inductance L e (x) mh inductance 1,00 0,75 0,50 0,25 voice coil 0,00-7,5-5,0-2,5 0,0 2,5 5,0 7,5 x[mm] L e (x) determined by geometry of coil, gap, magnet optimal size and position of short cut ring The power of Loudspeaker Models, 117th AES Convention, 26

14 Discussion Which nonlinearities are relevant and should be modelled? The power of Loudspeaker Models, 117th AES Convention, 27 Criteria for dominant Nonlinearities limits acoustical output generates audible distortion indicates an overload situation related with cost, weight, volume, efficiency The power of Loudspeaker Models, 117th AES Convention, 28

15 Road 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 The power of Loudspeaker Models, 117th AES Convention, 29 Parameter Measurement Particular unit Identification Sd Bl(x), Mms Kms(x) 2 dle ( x) d x dx Bl( x) + i i = Mms + Rms + Kms( x) x 2 dx dt dt dx d( Le ( x) i) u = Rei + Bl( x) + dt dt Differential Equation Valid for the particular unit The power of Loudspeaker Models, 117th AES Convention, 30

16 Method: Dynamical Measurement of Small Signal Parameters F tangent Apply small ac-stimulus x Measure state variables (voltage, current, displacement) Estimate optimal parameters (by fitting the linear model) K MS (x=0)= F AC x AC Perturbate loudspeaker to dispense with mechanical sensor The power of Loudspeaker Models, 117th AES Convention, 31 Small-Signal Signal 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? The power of Loudspeaker Models, 117th AES Convention, 32

17 Measure The Moving System Don t include the motor and chassis! The power of Loudspeaker Models, 117th AES Convention, 33 The Chassis Must Not Move OK with or without baffle if rigid The power of Loudspeaker Models, 117th AES Convention, 34

18 Axis Horizontal or Vertical? X g = 25 / f S2 cm (Olson) The power of Loudspeaker Models, 117th AES Convention, 35 What is a Small Signal? Some standards and specifications require one watt often not linear We know f S can vary with level (C MS is not a constant) Do not expect same result from different methods (different levels) Box design is still possible! The power of Loudspeaker Models, 117th AES Convention, 36

19 Voltage 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 The power of Loudspeaker Models, 117th AES Convention, 37 That Extra Measurement (1) Added Stiffness: account for altered mass V AS /V B = (f C2 /f S2 ) 1 (no mass change) V AS /V B = (f C Q EC /f S Q ES ) 1 (Thiele) (May be hard to assess added volume accurately) The power of Loudspeaker Models, 117th AES Convention, 38

20 That Extra Measurement (2) Added Mass: account for altered compliance M MS /M AM = {(f S2 /f AM2 ) 1} -1 (no compliance change) M MS /M AM = {(f S Q EAM /f AM Q ES ) 1} -1 (corrected for C shift) (Only valid if Bl does not change!) The power of Loudspeaker Models, 117th AES Convention, 39 That Extra Measurement (3) Added Luxury: with displacement laser, no disturbance to the driver all data can be collected at same time The power of Loudspeaker Models, 117th AES Convention, 40

21 Discussion Is it useful to apply a linear model to a loudspeaker operated in the large signal domain and to work with "effective" parameters? The power of Loudspeaker Models, 117th AES Convention, 41 Using a linear model in the large signal domain 2.5 -x_max < x < x_max force factor b(x) KLIPPEL x_max < x < x_max inductance L_E(x) KLIPPEL x_max < x < x_max stiffness K_MS(x) KLIPPEL b [N/A] 1.5 L_E [mh] K_MS [N/mm] << coil in x [mm] coil out >> << coil in x [mm] coil out >> << coil in x [mm] coil out >> Bl const. L e (xpeak) const. K ms const. PROBLEM: Effective Parameters depend on stimulus!! The power of Loudspeaker Models, 117th AES Convention, 42

22 Method: Static Large Signal Measurement F secant Sample the working range x Generate DC displacement x DC Measure associated state signals F DC K MS (x)= F DC x DC Calculate instantaneous parameters Repeat the measurement at other working points The power of Loudspeaker Models, 117th AES Convention, 43 Static measurement technique The power of Loudspeaker Models, 117th AES Convention, 44

23 Method: Quasi-static Measurement F DC FAC tangent Sample the working range Generate a variable offset X DC Excite with small AC-signal Measure state variables X AC and F AC Calculate gradient K grad (X DC ) Repeat measurement at other working points Parameter Transformation K grad (x)= F AC X AC x DC Transformation x AC K MS (x)= F X The power of Loudspeaker Models, 117th AES Convention, 45 Measuring Loudspeaker Excursion David Clark DLC Design The power of Loudspeaker Models, 117th AES Convention, 46

24 Music Excursion Picture The power of Loudspeaker Models, 117th AES Convention, 47 Drive-Unit Measurement at 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 The power of Loudspeaker Models, 117th AES Convention, 48

25 Pressure chamber Picture of DUMAX DUT Laser Computer Electronics unit I/O connections Pressure source Small adaptor boards The power of Loudspeaker Models, 117th AES Convention, 49 DUMAX spec Mfg., Model Sonavox XW7F AB # Outside Dimensions Max Diameter (mm): 204 Description: 5x7" Full Range Min Diameter (mm): 145 Mounting Depth (mm): 56 Sample # Sample #4 Magnet Diameter (mm): 73 Driver weight (Kg): 0.78 Report date: 03/14/03 Technician: Ponte Thiele-Small x=0 Electro-Mechanical x=0 Fs Hz Mmd 6.76 g By Fs Oscillation Qe 1.13 Cms m/n mm/n Qm 3.94 Rms 0.94 ohmm Qts 0.88 Sd m^2 effective radius Vas ltr. Bl 3.35 N/A #VALUE! mm Sd m^2 Re 3.42 ohm Re 3.42 ohm X Parameters from Curve Fit Xmag 2.7 mm Xsus 5.1 mm Xmax 2.7 Mag Maximum -0.2 mm Sus Minimum -0.2 mm Bl (N/A) Bl vs. X BL vs. x (measured) 71% of rest BL value (Xmag) Bl quadratic curve fit Kms (N/mm) Incremental Kms vs. X Kms vs x (measured) 400% (4X) of rest Kms ( Xsus) Kms 4th order curve fit displacement (mm) displacement (mm) The power of Loudspeaker Bl fit between Models, 117th AES Convention, 50 Kms fit between -3 mm and 2.5 mm -5 mm and 5.5 mm

26 DUMAX specs Mfg., Model Panasonic, A H Outside Dimensions Max Diameter (mm): 202 Description: 6.5" Poly Dual Cone Min Diameter (mm): 158 Mounting Depth (mm): 36 Sample # Mfg., Model 7Z800 Panasonic, A H Magnet Outside Diameter Dimensions (mm): 65.5 Max Driver Diameter weight (mm): (Kg): Report date: 10/21/03 Description: Diameter Technician: (mm): 158 Busch 6.5" Poly Dual Cone Min Mounting Depth (mm): 36 Sample # Z800 Magnet Diameter (mm): 65.5 Driver weight (Kg): 0.59 Thiele-Small Report date: Parameters x=0 Electro-Mechanical Technician: Busch x=0 Fs Hz Mmd 7.06 g By Fs Oscillation Qe 0.99 Cms m/n mm/n Thiele-Small x=0 Electro-Mechanical x=0 Qm Fs Hz Rms Mmd g ohmm By Fs Oscillation Qts 0.77 Sd Cms m/n m^2 effective mm/n radius Qe Qm 3.54 Vas ltr. Bl Rms ohmmn/a mm Sd m^2 Re Sd Bl m^2 ohm 2.39 N/A Qts 0.77 effective radius Re ohm Vas ltr mm Sd m^2 Re 1.85 ohm Re 1.85 ohm X Parameters from Curve Fit Xmag 2.7 mm X Xsus Parameters from Curve Fit 3.3 mm Xmax mm Xmax 2.7 Xmag Mag MaximumMag Maximum 0.2 mm Xsus Sus Sus Minimum Minimum 3.3 mm mm mm 0.2 mm Bl vs. X Incremental Kms vs. X Bl vs. X Incremental Kms vs. X Kms vs x (measured) 400% (4X) of rest Kms ( Xsus) Kms 4th order curve fit Kms vs x (measured) 400% (4X) of rest Kms ( Xsus) Kms 4th order curve fit Bl (N/A) Bl (N/A) BL vs. x (measured) % of rest BL value (Xmag) Bl Eff curve fit BL vs. x (measured) displacement (mm) displacement (mm) 71% of rest BL value (Xmag) Bl Eff curve fit Bl fit between Kms fit between mm and 5 mm -3 mm and 3 mm displacement (mm) displacement (mm) The power of Loudspeaker Models, 117th AES Convention, 51 Bl fit between Kms fit between Kms (N/mm) Kms (N/mm) DUMAX SIMULATED EXCURSION Begin The power of Loudspeaker Models, 117th AES Convention, 52

27 +10 mm The power of Loudspeaker Models, 117th AES Convention, mm The power of Loudspeaker Models, 117th AES Convention, 54

28 0 mm The power of Loudspeaker Models, 117th AES Convention, mm The power of Loudspeaker Models, 117th AES Convention, 56

29 -10 mm The power of Loudspeaker Models, 117th AES Convention, 57 Interpreting 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 The power of Loudspeaker Models, 117th AES Convention, 58

30 Dynamic Large Signal Measurement Method: Excite speaker with large AC-signal F AC (t) F secant x Measure state variables Estimate parameters to describe the relationship between state variables X AC (t) K MS (x)= F X The power of Loudspeaker Models, 117th AES Convention, 59 Large Signal Identification amplifier Distortion Analyzer gives electrical, mechanical, acoustical parameters Loudspaaker system 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 The power of Loudspeaker Models, 117th AES Convention, 60

31 Discussion Are systematic differences in the results of static, quasi-static or dynamic measurements? The power of Loudspeaker Models, 117th AES Convention, 61 Road 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 The power of Loudspeaker Models, 117th AES Convention, 62

32 Applications 1. Electrical Control of Loudspeaker System Model parameters Audio signal Targets: controller Equalization of the linear amplitude and phase response Linearization (compensation of distortion) Mechanical and thermal Protection On-line Diagnosis The power of Loudspeaker Models, 117th AES Convention, 63 Electrical Control of Loudspeakers Level & Trans. DRC Tra nsd uce r EQ Protection Nonlinear compensation DSP Tuning, diagnostic Power amplifier System measurement Transducer & acoustics Enable, in real-time: Equalisation Protection Nonlinear compensation Tuning & diagnostic The power of Loudspeaker Models, 117th AES Convention, 64

33 Model-based algorithms Generic 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. The power of Loudspeaker Models, 117th AES Convention, 65 Transducer Protection X-Prot/Therm X-Prot/Disp Temperature predictor Displacement predictor Input Temperature limitter Displacement limitter Output Protection based on real-time state prediction (temperature, displacement) Enables re-specification (de-rating) of loudspeaker input limits The power of Loudspeaker Models, 117th AES Convention, 66

34 Tuning & Diagnostic ut () Power amplifier i c shunt res. v c Plant Loudspea ker v c Adaptive filter (plant model) - Σ ε[ n] ic( t) ud( t) pt () 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 The power of Loudspeaker Models, 117th AES Convention, 67 Transducer nonlinearity compensation xd[ n] Linear dynamics Σ Σ 1/σ x 1/φ() x Σ R eb vc[ n] z -1 z -1 a 1 Σ k1( x) 1/φ() x a 2 x n d[ -1] Σ b dt 0 z -1 Σ ud[ n] a dt 1 b dt 1 Based on inverted loudspeaker model Frees motor design for sensitivity optimisation The power of Loudspeaker Models, 117th AES Convention, 68

35 Applications 2. Parameter Measurement Particular unit Identification Sd Bl(x), Mms Kms(x) 2 dle ( x) d x dx Bl( x) + i i = Mms + Rms + Kms( x) x 2 dx dt dt dx d( Le ( x) i) u = Rei + Bl( x) + dt dt Differential Equation Targets Specification of loudspeaker systems Interface between driver and system design Quality Control The power of Loudspeaker Models, 117th AES Convention, 69 Inter-company communication Supplier A Transducer A 1 Transducer A 2... Transducer A N Supplier B Transducer B 1 Transducer B 2... Transducer B N Supplier C Transducer C 1 Transducer C 2... Information management Integrator Product 1 Product 2... Product N Model provides a structure for intercompany information management Thiele-Small Parameters form a basis Simple extensions possible for Nonlinear characteristics Thermal behaviour symbol name category typ value unit f c loudspeaker cut-off freq. 950 Hz Q c loudspeaker Q-value loudspk. & cab. 8.5 m t total moving mass 6.00E-05 kg R eb DC-resist ance 7.2 Ohm φ 0 transduction coefficient (B l) Tm, Wb/m, N/A x lm m displacement limit m R tv voice-coil thermal res. 100 C/W C tv voice-coil thermal cap J/ C R tm magnet thermal res. 25 C/W C tm magnet thermal cap. 0.5 J/ C T lm m max. voice-coil temp. 100 C loudspk. φ 1 phi φ 2 phi E+06 φ 3 phi - 3 φ 4 phi - 4 k 1 k - 1 k 2 k - 2 k 3 k - 3 k 4 k - 4 Transducer C N The power of Loudspeaker Models, 117th AES Convention, 70

36 Del 75 ta Tv [K] Delta Tv P 4,0 3,5 KLIPPEL 3,0 2,5 P [W] 2,0 1,5 1,0 0, t [sec] force factor b(x) -x_max < x < x_max 5,5 KLIPPEL 5,0 4,5 4,0 3,5 b [N/ 3,0 A] 2,5 2,0 1,5 1,0 0,5 0,0-10,0-7,5-5,0-2,5 0,0 2,5 5,0 7,5 10,0 << coil in x [mm] coil out >> [V] 0,3 0,2 0,1 0,0-0,1-0,2-0,3-0,4-0,5 Stimulus (t) Stimulus (t) vs time Time [ms] Applications 3. Synthesis and Diagnosis of Loudspeakers Parameters Bl(x), Sd Mms Kms(x) Stimulus (Music) MODEL Output Symptoms (Distortion) Targets Prediction of the Transfer Behavior Relationship between causes and symptoms Investigation of design choices The power of Loudspeaker Models, 117th AES Convention, 71 Parameters are a common language between different tools Power Test thermal Symptoms Large signal behavior SIM Acoustical Measurement FEM motor design electrical Parameters acoustical mechanical Room Simulation Parameter measurement Linear Design Box, Cone, Crossover The power of Loudspeaker Models, 117th AES Convention, 72 Vibration (FEM) Radiation (BEM)

37 Applications 4. Auralization Targets: Objective Assessment with music Subjective Evaluation of sound quality Optimization of cost/performance ratio Tuning to the market The power of Loudspeaker Models, 117th AES Convention, 73 Auralization in Loudspeaker Development Development Manufacturing Marketing Management Objective Evaluation Distortion, Maximal Output Displacement, Temperature Evaluation of Design Choices Indications for Improvements Subjective Evaluation Personal Impression Sufficient Sound Quality Tuning to the target market Performance/Cost Ratio The power of Loudspeaker Models, 117th AES Convention, 74

38 Participate in Interactive Listening Test Parameters results Test signal Music AURA report MP3 files states distortion Server Inter net High quality judmements The power of Loudspeaker Models, 117th AES Convention, 75 Road 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 The power of Loudspeaker Models, 117th AES Convention, 76

39 Some unmodelled Phenomena 1. Other visco-elastic effects K(x=0, xpeak) 2. flux modulation Inductance L(x,i) Force factor Bl(x,i) The power of Loudspeaker Models, 117th AES Convention, 77 Dependency of Cms(f) on Frequency Magnitude of transfer function Hx(f)= X(f)/U(f) Measured Fitted without creep Creep model [mm/v] k 2k 5k 10k Frequency [Hz] f [Hz] The power of Loudspeaker Models, 117th AES Convention, 78

40 Considering Creep Dissipative model I R e Z L (jω) C 1 v R 2 M ms R ms Kundsen and Jensen, JAES 1993 U Blv Bl Bli C 2 Non-dissipative model R e C ms (f) M ms R ms I U Z L (jω) Blv Bl v Bli ( f ) = C ms 1 λ log Cms 10 f f s The power of Loudspeaker Models, 117th AES Convention, 79 Interaction between Nonlinearities Bl(x) Creep e.g. λ-parameter Cms(x) DC Force Cms(f=0) DC Displacement Le(x) The dc displacement may be increased by creep The power of Loudspeaker Models, 117th AES Convention, 80

41 Dependency of resonance on peak displacement resonance frequency (linear model) f s Hz Hz Hz 50 Small Signal Domain Large Signal Domain 27 % variation In small signal domain 40 0, Peak value of displacement mm The power of Loudspeaker Models, 117th AES Convention, 81 Dependency on Peak Displacement [N/mm] 3,0 2,5 2,0 1,5 1,0 voltage 0,5 0, Displacement x [mm] significant variation at the rest position x=0 The power of Loudspeaker Models, 117th AES Convention, 82

42 Mechanical properties of loudspeaker suspension parts Relation between force and deflection not one-to-one Here: two separately measured curves (for positive and negative direction) Hysteresis: relation between force and deflection dependent on previous history this points to visco-elastic behaviour How can we describe visco-elasticity? The power of Loudspeaker Models, 117th AES Convention, 83 1) Creep experiment: σ A certain stress σ 0 is applied for a certain time and the strain ε is measured σ 0 ε t Elastic: strain follows stress (Hooke) ε = J * σ 0 (J = compliance) t ε Visco-elastic: one part of the strain occurs instantaneously, another part while the stress is applied (creeping) t The power of Loudspeaker Models, 117th AES Convention, 84

43 1) Creep experiment: σ σ 0 Delivers creep compliance J(t) ε = σ 0 * J 0 t = t 0 J 0 = elastic compliance ε t ε(t) = σ 0 * J(t) t 0 < t < t 1 J(t) = creep compliance t 0 t 1 t The power of Loudspeaker Models, 117th AES Convention, 85 2) Relaxation experiment: ε A certain strain ε 0 is applied and the stress σ is measured ε 0 t Elastic: stress follows strain (Hooke) σ = E * ε (E = Young s modulus) σ t Visco-elastic: stress reaches an initial value and decreases subsequently (relaxation) σ t The power of Loudspeaker Models, 117th AES Convention, 86

44 2) Relaxation experiment: ε ε 0 Delivers relaxation modulus E(t) σ = σ 0 t = t 0 σ (t) = ε 0 * E(t) t 0 < t < t 1 E(t) = relaxation modulus σ σ 0 t t t 0 t 1 The power of Loudspeaker Models, 117th AES Convention, 87 Two visco-elastic phenomena: 1) Creep ε 2) Relaxation σ t 0 t 1 t t 0 t 1 t Can we observe these visco-elastic phenomena in loudspeaker suspension parts? The power of Loudspeaker Models, 117th AES Convention, 88

45 Creep measurement of a 80mm rubber surround: The power of Loudspeaker Models, 117th AES Convention, 89 Relaxation measurement of a 80mm rubber surround: The power of Loudspeaker Models, 117th AES Convention, 90

46 Influence of driving speed on hysteresis curve Green: 12mm/s Red: 2mm/s Blue: 0.2mm/s The power of Loudspeaker Models, 117th AES Convention, 91 The effects of viscoelastic behavior in suspension 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) The power of Loudspeaker Models, 117th AES Convention, 92

47 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 The power of Loudspeaker Models, 117th AES Convention, 93 What is flux 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) The power of Loudspeaker Models, 117th AES Convention, 94

48 Voice Coil Inductance L e (i) B Nonlinear relationship i=0 voice coil i=10 A i= -10 A H L e (i) depends on Material (permeability) and geometry of iron path voice coil height, number of windings The power of Loudspeaker Models, 117th AES Convention, 95 Flux Modulation by Bl(i,x) magnet permanent flux Φ 0 5,0 Force factor Bl(x,i) i=0 i= - 10 A i = 10 A 4,5 alternating flux Φ A (i) Voice coil Bl [N/A] 4,0 3,5 3,0 Pole piece Current i 2,5-10,0-7,5-5,0-2,5 0,0 2,5 5,0 7,5 10,0 Displacement X [mm] Linear Superposition of Φ A (i) and Φ 0 The power of Loudspeaker Models, 117th AES Convention, 96

49 Bl vs Current and Displacement 10 Flux modulation picture 8 Bl (N/amp) Current (amps.) displacement (mm) The power of Loudspeaker Models, 117th AES Convention, 97 Summary 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 The power of Loudspeaker Models, 117th AES Convention, 98

50 Do you know 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, 117th AES Convention, 99