Hot and Nonlinear Loudspeakers at High Amplitudes

Hot and Nonlinear Loudspeakers at High Amplitudes by Wolfgang Klippel Tutorial Presented at 131st AES Convention, New York, October 2011 Klippel, Tutorial: Loudspeakers at High Amplitudes, 1

Abstract: Nonlinearities inherent in electro-dynamical transducer and the heating of the voice coil and magnetic system limit the acoustical output, generate distortion and other symptoms at high amplitudes. The large signal performance is the result of a deterministic process and predictable by lumped parameter models comprising nonlinear and thermal elements. The tutorial gives an introduction into the fundamentals, shows alternative measurement techniques and discusses the relationship between the physical causes and symptoms depending on the properties of the particular stimulus (test signal, music). Selection of meaningful measurements, the interpretation of the results and practical loudspeaker diagnostic is the main objective of the tutorial, which is important for designing small and light transducers producing the desired output at high efficiency and reasonable cost. Klippel, Tutorial: Loudspeakers at High Amplitudes, 2

Our topic today 1. Large Signal Behavior (Symptoms) 2. Physical Causes and Models 3. Measurement of Large Signal Parameters 4. Loudspeaker Diagnostics 5. Modern Loudspeaker Design 6. Active Control 7. Conclusion Klippel, Tutorial: Loudspeakers at High Amplitudes, 3

Assessment of Sound Quality Amplitude Related to size, weight, cost!!! X [mm] 30 10 3 Nonlinear and Thermal Model Destruction Large signal performance Maximal Output Distortion Power Handling Stability Compression 1 0,3 Linear Model Small signal performance Bandwidth Sensitivity Flatness of Response Impulse Accuracy voice-coil displacement Klippel, Tutorial: Loudspeakers at High Amplitudes, 4

Watch for Nonlinear Symptoms! stroboscope Generator tone at f pointer scale Resonance frequency f s 1. Experiment f < f s 2. Experiment f f s 3. Experiment f > f s Klippel, Tutorial: Loudspeakers at High Amplitudes, 5

Vibration Behavior Klippel, Tutorial: Loudspeakers at High Amplitudes, 6 Klippel GmbH

1. Symptom: Distorted Waveform Input: sinousoidal voltage signal at 20 Hz Output: Voice coil Displacement Input signal Y2(t) vs time ZOOM Y2(t) KLIPPEL X [mm] 7,5 5,0 2,5 0,0 Distortion in the time domain -2,5-5,0 0,00 0,05 0,10 0,15 0,20 0,25 Time [s] Klippel, Tutorial: Loudspeakers at High Amplitudes, 7

2. Symptom: Harmonic Distortion Stimulus Response Amplitude Amplitude Distortion in the frequency domain DC harmonics f 1 frequency f 1 frequency A single tone generates harmonics and a DC component (in displacement) Klippel, Tutorial: Loudspeakers at High Amplitudes, 8

3. Symptom: Intermodulation Distortion Two-tone Stimulus Amplitude sound pressure spectrum harmonics difference tones summed tones f 1 bass tone f 2 voice tone frequency Klippel, Tutorial: Loudspeakers at High Amplitudes, 9

Example: Reproduced Two-tone Signal input Response 1 Frequency Domain Nonlinear System output Response 1 Frequency Domain dbu (Uo = 1V) 20 10 0-10 -20-30 -40-50 10 1 10 2 10 3 f [Hz] dbu (Uo = 1V) 20 10 0-10 -20-30 -40-50 10 1 10 2 10 3 f [Hz] Two-tone signal Fundamentals Harmonics Intermodulation Klippel, Tutorial: Loudspeakers at High Amplitudes, 10

4. Symptom: DC-Displacement Input signal Y2(t) vs time K MS (x) Y2(t) ZOOM KLIPPEL 7,5 5,0 X [mm] 2,5 0,0 DC displacement rest position of the coil x -2,5-5,0 0,00 0,05 0,10 0,15 0,20 0,25 Time [s] Dc displacement is generated dynamically by rectification of ac components Caused by asymmetrical nonlinearities working point is shifted away from rest position significant amplitude X DC (comparable with fundamental) usually in displacement (not in velocity, accelaration, input current) Klippel, Tutorial: Loudspeakers at High Amplitudes, 11

5. Symptom: Amplitude Compression Fundamental component X ( f1, U1 ) 23.4 Hz 2,5 Linear System 2,0 KLIPPEL X [mm] (rms) 1,5 1,0 0,5 0,0 0,0 2,5 5,0 7,5 10,0 12,5 15,0 Voltage U1 [V] Klippel, Tutorial: Loudspeakers at High Amplitudes, 12

Compression of SPL Output for a sinusoidal tone versus frequency Sound Pressure Response db - [V] (rms) Long Term Response linear response 130 KLIPPEL 125 120 115 110 105 100 95 Limited by peak displacement 90 Limited by coil temperature Long term response was measured by using a stepped sine wave and cycling 1 min on/1 min off 85 80 20 50 200 500 2k Frequency [Hz] Klippel, Tutorial: Loudspeakers at High Amplitudes, 13

Compression of 3 rd -order Harmonic Third-order harmonic distortion in percent (IEC 60268) Signal at IN1 80 0.50 V 1.57 V 2.64 V 3.71 V 4.79 V 5.86 V 6.93 V 8.00 V KLIPPEL 70 60 Voltage 50 Percent 40 30 20 10 0 4*10 1 6*10 1 8*10 1 10 2 Frequency f1 [Hz] Klippel, Tutorial: Loudspeakers at High Amplitudes, 14

Harmonic Distortion Second-order harmonic distortion in percent (IEC 60268) Signal at IN1 25 3.71 V 4.79 V 5.86 V 6.93 V 8.00 V KLIPPEL 20 Percent 15 10 5 4*10 1 6*10 1 8*10 1 10 2 Frequency f1 [Hz] Nonlinear Distortion depend on frequency and voltage Complicated amplitude characteristic (compression, reduction) Measurement versus amplitude also required (3D measurement) Klippel, Tutorial: Loudspeakers at High Amplitudes, 15

6. Variation of Resonance Frequency caused by suspension nonlinearity K(x) Amplitude response of fundamental component in displacement X ( f1, U1 ) 10 0 [mm] 10-1 6 db steps 10-2 resonance peak 10-3 10 100 200 Frequency [Hz] Klippel, Tutorial: Loudspeakers at High Amplitudes, 16

7. Symptom: Bifurcation into Multiple States X [mm] 6 5 Sweeping up 4 unstable states 3 2 1 backbone curve Sweeping down 10 100 frequency f [Hz] Small signal resonance Large signal resonance Klippel, Tutorial: Loudspeakers at High Amplitudes, 17

8. Symptom: Instabilities Unstable for f > f s Occurs if: - Driver has soft linear suspension - Equal-length configuration (Bl(x) nonlinearity) - Sinusoidal stimulus f > fs Bl(x) 5,0 4,5 [N/A] 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 KLIPPEL Paket bifurcation caused by motor.mov -5-4 -3-2 -1 0 1 2 3 4 5 [mm] x Bifurcation into two stable states of vibration Klippel, Tutorial: Loudspeakers at High Amplitudes, 18

9. Symptom: Chaotic Behavior x(t) Bl(x) 5,0 KLIPPEL 30 KLIPPEL 4,5 [N/A] 4,0 20 3,5 3,0 10 2,5 2,0 1,5 1,0 x [mm] 0-10 0,5-20 -5-4 -3-2 -1 0 1 2 3 4 5 [mm] x -30 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 t [s] Generation of subharmonics Stochastic responses for sinusoidal excitation How far away from chaos? Klippel, Tutorial: Loudspeakers at High Amplitudes, 19

What Limits the Acoustical Output? Limiting factors: 1. Maximal temperature of the voice coil 2. Maximal voice coil displacement (e.g. in suspension) 3. Maximal acceleration, forces (e.g. in the cone) 4. Maximal sound pressure (e.g. in the horn) Those factors depend on Properties of the transducer (linear, nonlinear and thermal parameters) Properties of the enclosure (sealed or vented box) Properties of the stimulus at the terminals (spectral and amplitude distribution, variation over time) Ambient condition (temperature, air convection, ) Klippel, Tutorial: Loudspeakers at High Amplitudes, 20

Maximal Sound Pressure Output for a sinusoidal input with different On/Off Cycle db - [V] (rms) 140 135 130 125 120 115 110 105 SOUND PRESSURE LEVEL Maximal SPL (short term) Cycled sine wave 1 s on, 1 min off KLIPPEL 100 95 90 Maximal SPL (long term) Cycled sine wave 1 min on, 1 min off 20 50 100 200 500 1k 2k 5k Limited by displacement Frequency [Hz] Limited by acceleration Limited by coil temperature Klippel, Tutorial: Loudspeakers at High Amplitudes, 21

Agenda 1. Large Signal Behavior (Symptoms) 2. Physical Causes and Models 3. Measurement of Large Signal Parameters 4. Loudspeaker Diagnostics 5. Modern Loudspeaker Design 6. Active Control 7. Conclusion Klippel, Tutorial: Loudspeakers at High Amplitudes, 22

Loudspeakers - a nonlinear and time-varying system Amplitude Working range MODELING X [mm] 30 Destruction 10 3 Large signal domain Nonlinear Model Thermal Model 1 0,3 Small signal domain Linear Model voice-coil displacement Klippel, Tutorial: Loudspeakers at High Amplitudes, 23

How to Structure the Model? Domains and interfaces passed by the audio signal Input power Coil temperature Heat transfer Convection cooling Time constants thermal velocity Sound pressure in ear channel digital electrical mechanical acoustical psychoacoustical Data compression Sample rate conversion, AD conversion Amplifier Crossover Motor Suspension Cone Radiation, Diffraction Propagation, Room Influence Perception, Evaluation Voltage Coil displacement Cone displacement Klippel, Tutorial: Loudspeakers at High Amplitudes, 24

Power Distributed by the Loudspeaker Power density: 10 W / cm 3 1 W / cm 3 1 W / m 3 1 mw / m 3 Radiation Sound Propagation Room Acoustics p(r 1 ) Audio signal Amplifier Crossover EQ u(t) Electromechanical Transducer x(t) Mechanoacoustical Transducer (Cone) Radiation Sound Propagation Room Interference p(r 2 ) sound field linear i(t) nonlinear Radiation Sound Propagation Room Interference p(r 3 ) linear Klippel, Tutorial: Loudspeakers at High Amplitudes, 25

Stiffness K ms (x) of Suspension K 6 N/mm 5 total suspension F F x 4 3 x 2 1 spider surround -10.0-7.5-5.0-2.5 0.0 2.5 5.0 7.5 10.0 diplacement x mm restoring force F Kms ( x) x displacement Kms(x) determined by suspension geometry impregnation adjustment of spider and surround Klippel, Tutorial: Loudspeakers at High Amplitudes, 26

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Symmetrical Limiting of Spider stiffness K_MS(x) 4,0 -x_max < x < x_max KLIPPEL 3,5 3,0 diaphragm K_MS [N/mm] 2,5 2,0 1,5 frame 1,0 0,5 0,0-10 -5 0 5 10 << coil in x [mm] coil out >> spider voice coil former Requirements: geometry of inner corrugation roll is important sufficient numbers of corrugation rolls number of grooves equals numbers of ridges symmetrical feet Klippel, Tutorial: Loudspeakers at High Amplitudes, 28

Distortion Generated by K ms (x) simplified signal flow chart Voltage distortion fs highpass pressure pass band Displacement x fs lowpass Bass tone multiplier K ( x) K Bl ms ms (0) R e Multiplication of displacement time signals x(t)*kms(x(t)) Klippel, Tutorial: Loudspeakers at High Amplitudes, 29

Symptom of Kms: Amplitude Compression Fundamental coil displacement in a vented-box System Fundamental component X ( f1, U1 ) 1.00 V 2.00 V 3.00 V linear 6 4.00 V LINEAR MODEL 5 nonlinear X [mm] (rms) 4 3 2 Compression 1 0 10 1 10 2 Frequency f1 [Hz] Klippel, Tutorial: Loudspeakers at High Amplitudes, 30

Nonlinear Stiffness K ms (x) Symptoms: - Compression of the fundamental (f<2f s ) - Harmonic distortion (f<2f s ) - dc displacement (f 2f s ) Remedies: 1. Remove asymmetry in K ms (x) by using symmetrical geometry using a soft surround (spider dominant) compensating surround by spider asymmetries finding optimal working point in the surround 2. Reduce geometrical limiting by increasing number and size of corrugation rolls 3. Avoid loss of stiffness at x=0 by using material with low visco elasticity (low creep) Klippel, Tutorial: Loudspeakers at High Amplitudes, 31

F Bli F Force Factor Bl(x) N S back plate magnet pole plate Bl(x) determined by i Φ dc Magnetic field distribution B-field coil Height and overhang of the coil Optimal voice coil position F F Bl( x) i U Bl( x) v pole piece displacement 0 mm x Electro-dynamical driving force Voice coil current Back EMF Voice coil velocity 5.0 N/A 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Bl(x) -7.5-5.0-2.5 0.0 2.5 5.0 7.5 << Coil in X mm coil out >> Klippel, Tutorial: Loudspeakers at High Amplitudes, 32

Motor with Equal-length Configuration Coil height h coil gap height h gap force factor b(x) h coil 3,5 -x_max < x < x_max KLIPPEL magnet h gap pole plate voice coil 3,0 50% b [N/A] 2,5 2,0 1,5 h coil 1,0 0,5 pole piece x Properties: 0,0-7,5-5,0-2,5 0,0 2,5 5,0 7,5 << coil in x [mm] coil out >> Sensitive to offset in rest position Sensitive to instabilities f>f s Low-order distortion at low amplitudes Low inductance and flux modulation Klippel, Tutorial: Loudspeakers at High Amplitudes, 33

Motor with Overhang Coil Coil height h coil > gap height h gap h coil force factor b(x) magnet h gap pole plate -x_max < x < x_max 5,5 5,0 4,5 50% 4,0 h coil h gap KLIPPEL voice coil b [N/A] 3,5 3,0 2,5 2,0 h coil 1,5 pole piece 1,0 0,5 x Properties: 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 >> Insensitive to offset at rest position Low distortion for x < (h coil -h gap ) High distortion for x > (h coil -h gap ) High voice coil inductance Sensitive to flux modulation Klippel, Tutorial: Loudspeakers at High Amplitudes, 34

Adjusting Coil s Rest Position magnet pole plate magnet pole plate Induction B voice coil Induction B voice coil pole piece x=0 displacement pole piece x=x b displacement Force factor Bl vs. displacement X Force factor Bl vs. displacement X 5,0 Bl(X) 5,0 Bl(X) 4,5 4,5 4,0 4,0 3,5 3,5 3,0 3,0 Bl [N/A] 2,5 2,0 Bl [N/A] 2,5 2,0 1,5 1,5 1,0 1,0 0,5 0,5 0,0-10,0-7,5-5,0-2,5 0,0 2,5 5,0 7,5 10,0 Displacement X [mm] 0,0-10,0-7,5-5,0-2,5 0,0 2,5 5,0 7,5 10,0 Displacement X [mm] Klippel, Tutorial: Loudspeakers at High Amplitudes, 35

Generation of Bl(x) Distortion 1 st nonlinear effect: Parametrical Excitation pass band Voltage distortion impedance fs highpass pressure current fs Voice tone multiplier x lowpass fs Bass tone 1. Motor force F=Bl(x)*i 2. Multiplication of displacement x(t) and current i(t) 3. High distortion (f 1 f s, f 2 > f s ) Klippel, Tutorial: Loudspeakers at High Amplitudes, 36

Amplitude Modulation two-tone stimulus f 1 < f s, f 2 > f s Bl(x) 5,0 [N/A] 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 Symmetrical Force factor Bl(x) KLIPPEL -5-4 -3-2 -1 0 1 2 3 4 5 [mm] x Pfar [ N / m^2 ] 5,0 2,5 0,0-2,5 Sound pressure Pfar(t) in far field vs time Pfar(t) Rest position peak Mean Bottom -5,0 Cycle 0,05 0,10 0,15 0,20 0,25 0,30 Time [s] Klippel, Tutorial: Loudspeakers at High Amplitudes, 37

Symptoms of Bl(x) Intermodulation Distortion Third-order intermodulation distortion in percent (IEC 60268) Pressure Pfar in far field 125 intermodulation in same order of magnitude as harmonics for f < fs U = 1.5 U = 3 U = 4.5 U = 6 U = 7.5 U = 9 U = 10.5 U = 12 U = 13.5 U = 15 KLIPPEL magnet pole plate 100 75 Induction B voice coil 50 pole piece x=x b displacement 25 4*10 2 6*10 2 8*10 2 10 3 Frequency [Hz] (constant bass tone f 1 =0.5f s and varying voice tone). Parametric excitation F=Bl(x)*i Klippel, Tutorial: Loudspeakers at High Amplitudes, 38

Symptoms of Bl(x) dc displacement X dc 5 mm - 0 f s unstable 2f s frequency UNIQUE SYMPTOM Bl-maximum dc-displacement (X dc ) for f < f s : small X dc towards maximum of Bl-curve for f = f s (resonance): no dc-part generated (X dc = 0) for f > f s : X dc away from Bl-maximum for f 1.5fs: high values of X dc ( may become unstable) Klippel, Tutorial: Loudspeakers at High Amplitudes, 39

Instability of the Nonlinear Motor 145.0 Fundamental Pfar( f1, U1 ) 0.75 b(x) 5,0 Unstable for f > f s KLIPPEL 20 15 0.50 0.25 0.00 500 Pfar [N/m^2] U [V] 4,5 [N/A] 4,0 10 300 400 3,5 3,0 100 200 f [Hz] 2,5 5 2,0 1,5 1,0 0,5-5 -4-3 -2-1 0 1 2 3 4 5 [mm] x Effects: Reduced acoustical output Substantial Distortion Klippel, Tutorial: Loudspeakers at High Amplitudes, 40

Effect of a Nonlinear Motor Bl [N/A] 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 Curve 1 KLIPPEL driver with dominant Bl(x)-nonlinearity linear suspension constant inductance 0,0-15 -10-5 0 5 10 15 Displacement X [mm] Fundamental component X ( f1, U1 ) Parametric excitation F=Bl(x)*i mm 8 7 6 5 U = 1.5 U = 3 U = 4.5 U = 6 U = 7.5 U = 9 U = 10.5 U = 12 U = 13.5 U = 15 KLIPPEL displacement Nonlinear damping F=Bl(x) 2 /Re*v 4 3 2 1 0 10 2 10 3 Frequency [Hz] Klippel, Tutorial: Loudspeakers at High Amplitudes, 41

F Bli F Force Factor Bl(x) Symptoms: - Compression of the fundamental (f<f s ) - Enhancement of the fundamental (f f s ) - Harmonic distortion (f<2f s ) - Intermodulation distortion (f 1 < f s, f 2 > f s ) - dc displacement (f 2f s ) Remedies: 1. Reduce Bl(x) asymmetries by placing coil at optimal rest position by using a symmetrical B-field in the gap 2. Make Bl(x)=constant by increasing voice coil overhang by increasing voice coil underhang N i S Klippel, Tutorial: Loudspeakers at High Amplitudes, 42

Voice Coil Inductance L e (x) Φ coil (-9 mm) Φ coil (+9 mm) 4.0 Le [mh] 2.5 2.0 Without without shorting rings With with shorting rings Φ counter 1.5 1.0 0.5 shorting ring voice coil displacement -9 mm 0 mm 9 mm x 0.0-15 -10-5 0 5 10 15 << Coil in X [mm] coil out d ( x, i) d L( x) i U ind dt dt Reluctance force 2 i ( t) dl( x) F rel 2 dx Differentiated Magnetic flux L e (x) determined by geometry of coil, gap, magnet optimal size and position of short cut ring Klippel, Tutorial: Loudspeakers at High Amplitudes, 43

Effect of Nonlinear Inductance L e (x) [mh 0,30 0,25 0,20 Le(X) KLIPPEL Le(x) nonlinearity causes variation of electrical input impedance 0,15 0,10 0,05 0,00-4 -3-2 -1-0 1 2 3 4 X=-4 mm << Coil in X [mm] coil out >> X=4 mm Magnitude of electric impedance Z(f) 25 x= 0 mm x = - 4 mm x = + 4 mm KLIPPEL 20 [Ohm] 15 10 5 0 X=-4 mm Coil is clamped X=4 mm 10 1 10 2 10 3 10 4 Frequency [Hz] Klippel, Tutorial: Loudspeakers at High Amplitudes, 44

Distortion caused by L e (x) pass band Voltage distortion differentiator i impedance fs highpass pressure 6dB/oct multiplier L( x) 1 L(x) L (0) current x displacement fs fs lowpass Voice tone f 2 Bass tone f 1 1. Multiplication of x(t) and i(t) 2. Differentiation of distortion + High-pass filtering High intermodulation distortion (f 1 f s, f 2 > f s ) Klippel, Tutorial: Loudspeakers at High Amplitudes, 45

Remedies for L e (x) Symptoms: - Intermodulation distortion (f 1 < f s, f 2 > f s ) - dc displacement (f>f s ) Remedies: 1. Reduce magnetic ac-flux by using a smaller coil with less windings by increasing the voice coil resistance Using shorting material 2. Make Le(x)=constant By placing shorting material at right position Klippel, Tutorial: Loudspeakers at High Amplitudes, 46

Remedy for L e (x): Shorting Ring R RING magnet pole plate L 2 (x) Coil flux A R E (T V ) L E (x) C MS (x) M MS R MS F m (x,i) Counter flux C voice coil short cut ring i u R 2 (x) b(x)v b(x) v b(x)i pole piece Electro-mechanical Equivalent Circuit L E (x) 1,75 1,50 1,25 -x_max < x < x_max inductance L_E(x) KLIPPEL Reduced value Effect depends on L_E [mh] 1,00 0,75 0,50 Geometry (Ring or Cap) Material (Aluminum or Copper) 0,25 0,00 Less variation versus x -5,0-2,5 0,0 2,5 5,0 << coil in x [mm] coil out >> x Size, position, distance to the coil Klippel, Tutorial: Loudspeakers at High Amplitudes, 47

Nonlinearities of Loudspeaker Ports Air plug Air plug Usual modeling the air flow by a lumped mass (air plug) is a simplification! jet stream in loudspeaker ports.mov Klippel, Tutorial: Loudspeakers at High Amplitudes, 48

Flow Resistance R A (v) of a Port at Medium Amplitudes V < V Crit R A (v) Generating an air jet energy dissipated in the far field Harmonics at low frequencies R A (v) ~ v *m v Klippel, Tutorial: Loudspeakers at High Amplitudes, 49

Asymmetrical Flow Resistance R A (v) p 0 p box p box x DC x DC R A (v 0 ) > R A (-v 0 ) R A (v) DC component in pressure p box dynamical voice coil offset x v Klippel, Tutorial: Loudspeakers at High Amplitudes, 50

Vented Box System Symptoms: Q-factor of port decreases with amplitude (port closes!!) Harmonics (not critical!!) Generates modulated air noise (critical!!) dc component in sound pressure in enclosure and a dynamic offest in voice coil working point (critical!!) Remedies: Keep velocity in port low (< 10 m/s) Keep port geometry symmetrical!! Klippel, Tutorial: Loudspeakers at High Amplitudes, 51

Nonlinear Sound Propagation Wave Steepening Speed of sound depends on sound pressure c(p) Sound pressure maxima travel faster than minima Klippel, Tutorial: Loudspeakers at High Amplitudes, 52

Wave Steepening in Horns Symptoms: dominant second order harmonic and intermodulation distortion Distortion rises by 6dB/oct. to higher frequencies Distortion rises with distance from throat Remedies: Reduce propagation where sound pressure is high High Flare rate of the horn Short horn length Low pressure at throat Klippel, Tutorial: Loudspeakers at High Amplitudes, 53

Nonlinear Mechanical Resistance R ms (v) v R ms (v) Air flow dome spider magnet gap v Pole piece R ms (v) depends on velocity v of the coil due to air flow and turbulences at vents and porous material (spider, diaphragm) Klippel, Tutorial: Loudspeakers at High Amplitudes, 54

Remedies for Nonlinear Mechanical Damping Problem: Air Flow in vents, holes, leakages or porous material in enclosures Dissipation depends on velocity (turbulences, energy is moved into the far field) Occurs in: Microspeaker, Headphones, Tweeter Horn compression driver Remedy: Increase electrical damping by increasing Bl or decreasing Re better sealing of the enclosure Klippel, Tutorial: Loudspeakers at High Amplitudes, 55

Which nonlinearities are important? Criteria for dominant Nonlinearities: limits acoustical output generates audible distortion, indicates an overload situation causes unstable behavior related with cost, weight, volume, efficiency affects speaker system alignment Klippel, Tutorial: Loudspeakers at High Amplitudes, 56

Ranking List of Transducer Nonlinearities 1. Force Factor Bl(x) 2. Compliance C ms (x) tweeter 3. Inductance L e (x) 4. Flux Modulation of L e (i) 5. Mechanical Resistance R ms (v) 6. Nonlinear Sound Propagation c(p) 7. Doppler Distortion (x) 8. Flux Modulation of Bl(i) 9. Nonlinear Cone Vibration 10. Port Nonlinearity R A (v) 11. many others... microspeaker woofers microspeaker horns Klippel, Tutorial: Loudspeakers at High Amplitudes, 57

Electro-mechanical Equivalent Circuit Nonlinear parameters: Force factor Bl(x) of the motor Compliance C MS (x, t) of the suspension Voice coil inductance L E (x), L 2 (x) resistance R 2 (x) due to eddy currents DC-resistance R E (T V ) Reluctance force F M (x) Compliance C r (p rear ) of rear enclosure Compliance C ab (p box ) of vented enclosure Losses in port R ap (q p ) Time delay t(x) due to Doppler effect are not constant parameters but depend on state variables: Displacement x Voice coil temperature T V Time t due to ageing volume velocity q p in port pressure p rear rear enclosure pressure p box in vented enclosure L 2 (x) R e (T v ) L e (x) F m (x,i) M ms R ms (v) C ms (x) S d V q p i R 2 (x) V p box R ap (q p ) U Bl(x,I)V Bl(x,I) Bl(x,I)I S d C ab (p box ) R al radiation p rear M ap C r (p rear ) Klippel, Tutorial: Loudspeakers at High Amplitudes, 58

Signal Flow Chart of the Transducer simplified by assuming a linear impedance for the mechanical load Feedback Linear system modeled by transfer function H(f,r a ) Nonlinear differential equation Based on lumped parameter modeling Feed-forward Klippel, Tutorial: Loudspeakers at High Amplitudes, 59

Source of the Nonlinear Distortion considering dominant nonlinearities in electrodynamical loudspeakers Loudspeaker u input u D Nonlinear System H(s,r 1 ) H(s,r i ) p(r i ) H(s,r N ) p(r 1 ) sound...... field............ p(r N ) Nonlinear distortion Klippel, Tutorial: Loudspeakers at High Amplitudes, 60

Voice Coil Heating depends on the Spectral Properties of the Stimulus 100 KLIPPEL 30 90 t 80 1 t 2 t 3 25 70 [K ] [W ] Music: T v /P re = 50 40 30 20 10 0-1 0 0 250 500 750 1000 1250 1500 1750 2000 2250 Classic Pop t [s e c ] Vocal 6,8 K/W 4,6 K/W 7,5 K/W P P Re RE T V Delta Tv 15 10 Thermal resistance is not constant!! 5 0 High voice coil displacement gives high Convection cooling Klippel, Tutorial: Loudspeakers at High Amplitudes, 61

Nonlinear Thermal Model R tv P tv P g R tg P mag T v P con R tc (v) R tt (v) T g T m P coil P eg T v C tv Air convection cooling R ta (x) C ta Direct heat transfer T g C tg T m C tm R tm T a Klippel, Tutorial: Loudspeakers at High Amplitudes, 62

Identification of the Driver Model Abstraction L 2 (x) i R E (T V ) u L E (x) R 2 (x) b(x)v b(x) v b(x)i C MS (x) M MS R MS F m (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) dt dt Lumped Parameter Model Differential Equation Identification Three Tasks: prove that the model is adequate for the particular driver measure the free parameters (C ms, Bl, M ms,...) of the model measure instantaneous state variables (displacement x,...) Klippel, Tutorial: Loudspeakers at High Amplitudes, 64

Large Signal Model & Parameters Heat transfer Convection cooling R tv, R tm, Thermal Model tv, tm, Stimulus Linear System H el (s) Nonlinear System Linear System H a (s) Sound pressure output Bl(x), L(x), Z m (s) L(i) Kms(x) T/S Parameter Amplifier Crossover Motor Suspension Cone Enclosure, Horn room Klippel, Tutorial: Loudspeakers at High Amplitudes, 65

Measurement of Large Signal Parameters Standard IEC 62458 defines 1. Static (quasi-static) method 2. Incremental dynamic method 3. Full dynamic method Klippel, Tutorial: Loudspeakers at High Amplitudes, 66

Method: Static Measurement F secant 1. Select a working point x 2. Excite with DC stimulus 3. Measure associated state signals K MS (x)= F DC x DC 4. Calculate instantaneous parameters 5. Repeat the measurement at other working points Klippel, Tutorial: Loudspeakers at High Amplitudes, 67

Example: Static Measurement of Stiffness Ringlstetter Harman Becker Straubing 2004 Klippel, Tutorial: Loudspeakers at High Amplitudes, 68

Incremental dynamic Measurement tangent Method: F DC F AC 1. Select a working point 2. Excite with a dc stimulus 3. Add a small AC-signal 4. Measure state variables X AC and F AC 5. Calculate gradient K grad (X DC ) 6. Repeat at other working points 7. Parameter Transformation K grad (x)= F AC X AC x DC Transformation x AC K MS (x)= F X Klippel, Tutorial: Loudspeakers at High Amplitudes, 69

First available product: DUMAX incremental dynamic method Photo courtesy by D. Clark Klippel, Tutorial: Loudspeakers at High Amplitudes, 70

Full dynamic Measurement F secant Method: 1. Excite speaker with audio-like signal F AC (t) x 2. Measure instantaneous state variables 3. Estimate free model parameters X AC (t) first application of system identification to loudspeakers K MS (x)= F X (Knudsen 1993) Klippel, Tutorial: Loudspeakers at High Amplitudes, 71

Full Dynamic Measurement of Loudspeaker Nonlinearities Suspension Part Measurement (SPM) Large Signal Identification (LSI) Long-term Power Testing (PWT) Motor-Suspension Check QC (MSC) Advantages: Loudspeaker under normal working conditions Audio-like stimulus On-line measurement Disadvantage: Large Signal Idenification requires nonlinear signal processing Klippel, Tutorial: Loudspeakers at High Amplitudes, 72

Adaptive Identification Principle full dynamic method based on current & voltage measurement audio-like, persistent Signal Source normal working conditions precise, robust sensor State Measurement Parameter Estimation Detection working range Model protection optimal fitting Klippel, Tutorial: Loudspeakers at High Amplitudes, 73

Dynamic Measurement of Motor and Suspension Nonlinearities Stimulus Noise, Audio signals (music, noise) Multi-tone complex Voltage & current Nonlinear System Identification State Variables peak displacement during measurement voice coil temperature eletrical input power, Linear Parameters T/S parameters at x=0 Box parameters fb,qb Impedance at x=0 Nonlinear Parameters nonlinearities Bl(x), Kms(x), Cms(x), Rms(v), L(x), L(i) Voice coil offset Suspension asymmetry Maximal peak displacement (Xmax) Thermal Parameters Thermal resistances Rtv, Rtm Thermal capacity Ctv, Ctm Air convection cooling Klippel, Tutorial: Loudspeakers at High Amplitudes, 74

Controlling the Production Process Shift coil 0.6 mm to backplate Action Fail Targets: Corrected Rest Position Failure: Coil Offset Detection of motor and suspension problem as fast as possible (when first device arrives at the end of line) B field voice coil B field voice coil Simplify interpretation of results (voice coil offset in mm) Single-valued parameter can be directly be used for adjustment Klippel, Tutorial: Loudspeakers at High Amplitudes, 75

Measurement of Thermal Parameters within the Large Identification Module (LSI) 20 10 Xpeak Voice coil displacement Xbottom KLIPPEL [mm] 0-10 -20 Delta Tv [K] 0 Delta 2500 Tv 5000 7500 P real 10000 12500P Re 15000 125 t [sec] KLIPPEL 100 75 50 25 0 r V R TM, M 0 2500 5000 7500 10000 12500 15000 t [sec] R TV, V nonlinear Thermal identification Voice coil temperature Real input power Power dissepated in Re Klippel, Tutorial: Loudspeakers at High Amplitudes, 76 225 200 175 150 125 100 75 50 25 0 P [W]

Small Signal Performance Specifications for Active and Passive Loudspeaker Systems 90 180 90 270 0 All information are provided by a complex 3D far field response H(f,, ) measured with sufficient angular resolution -90 Most important responses: SPL on-axis amplitude response under anechoic conditions (IEC 60268-5) Directivity Directivity index D i (f) or sound power response P a (f) (IEC 60268-5 Sec. 22.1) Group delay latency variation vs. Frequency, variation between channels Klippel, Tutorial: Loudspeakers at High Amplitudes, 78

Large Signal Performance Specifications for Active and Passive Loudspeaker Systems Maximal SPL max at 1 m, on-axis anechoic conditions, in frequency range Effective frequency range (Upper and lower limits f lower,l < f < f upper,l ) Flatness of on-axis response (maximal deviation of SPL on-axis response from mean SPL) Harmonic distortion (Equivalent input distortion) Intermodulation distortion (voice and bass sweep) Impulsive distortion (peak, crest) indicating rub&buzz, loose particles Modulated noise (MOD) indicating air leakage Durability verified in accelerated life test Klippel, Tutorial: Loudspeakers at High Amplitudes, 79

How to interprete Harmonic Distortion 2 nd, 3 rd and higher-order harmonics (amplitude and phase information) Important for irregular defects rub & buzz 2 nd harmonic 3 rd harmonic THD CHD Crest factor asymmetry symmetry magnitude smoothness Characteristics of the nonlinearity Klippel, Tutorial: Loudspeakers at High Amplitudes, 80

Interpretation of THD in SPL 130 120 Fundamental Fundamental Fundamental THD KLIPPEL db - [V] (rms) 110 100 90 80 70 60 THD 50 50 100 200 500 1k 2k 5k Frequency [Hz] Kms(x) Bl(x) L(x) L(i) Cone Vibration Klippel, Tutorial: Loudspeakers at High Amplitudes, 81

Transform Distortion to the Source Sinusoidal sweep H(f,r 1 ) p(r 1 ) U(f) sound field Distortion in Voltage D Nonlinear System H(f,r 2 ) p(r 2 ) Sound pressure measurement 3rd harmonic distortion in voltage Signal at IN1 3rd harmonics absolute Signal at IN1 0-5 nearfield 30 cm 60 cm 1 m distance KLIPPEL 90 85 1 m distance 60 cm distance 30 cm distance nearfield KLIPPEL -10-15 Independent of room 80 75 db - [V] -20-25 -30 Inverse filtering with H(f,r) db - [V] 70 65 60-35 55-40 50-45 45 50 100 200 500 1k Frequency [Hz] 40 50 100 200 500 1k Frequency [Hz] Klippel, Tutorial: Loudspeakers at High Amplitudes, 82

Interpretation of Multi-tone Distortion 120 110 Fundamental Fundamental Multi-tone Distortion KLIPPEL 100 90 [db] 80 70 60 50 50 100 200 500 1k 2k 5k 10k Frequency [Hz] Distortion Kms(x) L(x) Bl(x) L(i) Doppler Effect Cone Vibration Klippel, Tutorial: Loudspeakers at High Amplitudes, 83

How to cope with nonlinearities Measure nonlinear distortion in the near field ensure sufficient SNR Transform distortion to the loudspeaker input concept of equivalent input distortion Be aware of interactions between nonlinearities no compensation of Kms(x), Bl(x), L(x) Check for dc-displacement instability Use numerical simulation tool predict THD, Xmax, SPLmax, IMD, Pmax, T Separate regular nonlinearities from irregular defects measure Crest Factor of Distortion Klippel, Tutorial: Loudspeakers at High Amplitudes, 84

Thermal Parameters alpha 0.375914 Heating of voice coil by eddy currents Rtv 0.932822 K/W thermal resistance coil ==> pole tips rv 0.192945 Ws/Km air convection cooling depending on velocity Rtm 0.372579 K/W thermal resistance magnet ==> environment tau m 67 min thermal time constamt of magnet Ctm 10796.289063 Ws/K thermal capacity of the magnet tau v 131.063660 s thermal time constant of voice coil Ctv 140.502304 Ws/K thermal capacity of the voice coil Can be directly be pasted into the Simulation module (SIM) to make a thermal analysis (predict temperatures and power flow) Klippel, Tutorial: Loudspeakers at High Amplitudes, 85

Thermal Analysis single tone at 65 Hz with 40 V rms R tv P g T m T v P tv 0.93 K/ W P con P coil P RE + 0.3 (P real P RE ) T v C tv Air convection cooling R tc (v) P eg 0.7 (P real P RE ) Direct heat transfer T m C tm 0.37 K/ W R tm 1/r tv V rms T a Klippel, Tutorial: Loudspeakers at High Amplitudes, 86

Thermal Analysis single tone at 65 Hz with 40 V rms 146 W R 196 K tv P g 163 W T m 60 K T v P tv P con P coil 167 W T v C tv Air convection cooling R tc (v) Direct heat transfer P eg 17 W T m C tm R tm 21 W T a Klippel, Tutorial: Loudspeakers at High Amplitudes, 87

Thermal Analysis single tone at 1000 Hz with 40 V rms 69 W R tv P g 91 W T m T v 99 K 34 K P tv 70 W P coil P con 1 W P eg 22 W C tv C tm R tm T v R tc (v) Direct heat transfer T m T a Klippel, Tutorial: Loudspeakers at High Amplitudes, 88

Thermal Analysis two-tone at 1000 Hz and 50 Hz with 30 V rms 118 W R tv P g 134 W T m T v 160 K 50 K P tv P con P coil 134 W 16 W P eg C tv 16 W C tm R tm T v R tc (v) Direct heat transfer T m T a Klippel, Tutorial: Loudspeakers at High Amplitudes, 89

Power Bypass Factor The power P tv should be minimal! P coil T v T v C tv P tv P con R tc (v) 118 W 16 W R tv 16 W P eg P mag T m T m C tm R tm The bypass factor should be maximal! P P con P P eg Re R2 Total input power T a Klippel, Tutorial: Loudspeakers at High Amplitudes, 90

Optimal Thermal Design Investigation of Design Choices dome dome gap gap vent vent A v A v Vented pole piece Sealed pole piece Which design Choice provides a better heat transfer? Klippel, Tutorial: Loudspeakers at High Amplitudes, 91

Bypass Power Factor (vented and sealed pole piece) 60 dome 50 40 [%] 30 vent sealed vent open gap 20 vent 10 0 A v 1 10 100 1000 frequency [Hz] Closing the vent in the pole piece: 50 percent of the power will bypass the coil! Klippel, Tutorial: Loudspeakers at High Amplitudes, 92

Steps in Loudspeaker System Design 1. Definition of target performance and constraints 2. Defining the interface between the components (DSP, amplifier, transducers) 3. Specification of the components 4. Selecting the components 5. Building the first prototype 6. Verification of the performance Klippel, Tutorial: Loudspeakers at High Amplitudes, 94

Loudspeaker Development Parts (cone, spider, motor parts) Driver (woofer, tweeter) System (driver + Xover + room) Listener Modeling Distributed Parameter Models (FEA, BEA) Lumped Parameter Models System-oriented Models Psychoacoustical Models Measurement Application Geometry Material Parameter Thermal Parameters Linear T/S Parameters Cone Vibration Parameters independent of stimulus Nonlinear Parameters Definition of Target Performance Selection of Components & Design Evaluation of the First Prototype End-of-line Testing Nonlinear Distortion Power Handling Loudness density Small Signal Performance Symptoms dependent on stimulus Klippel, Tutorial: Loudspeakers at High Amplitudes, 95 R&D Sound Attributes Quality Metrics QC

How to define the Target Performance Auralization using the large signal model 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 Klippel, Tutorial: Loudspeakers at High Amplitudes, 96

Auralization - Systematic Listening Tests using Simulation and Decomposition Techniques Thermal Parameters R tv, R tm varied parameter Thermal Model Power Temperature Listening test Stimulus Linear System Nonlinear System Linear System Sensations H el (s) Transfer function Lumped Parameters Bl(x), L(x). Cms(x) Re, Mms Y(s) Mechancial Cone Admittance H a (s) Electrical Transfer Function Psychoacoustical Model Amplifier Crossover Motor Suspension Cone Enclosure, Horn Room Klippel, Tutorial: Loudspeakers at High Amplitudes, 97

Nonlinear Auralization Technique -Xprot < X < Xprot Xp- < X KLIPPEL < Xp+ 6 5 4 3 2 Stiffness of suspension Kms (X) Force factor Bl (X) -Xprot < X < Xprot Xp- < X < Xp+ 2,25 KLIPPEL 2,00 1,75 1,50 1 Parameters 0-7,5-5,0-2,5 0,0 2,5 5,0 7,5 1,25 X [mm] 1,00 0,75 0,50 0,25 0,00-7,5-5,0-2,5 0,0 2,5 5,0 7,5 X [mm] Bl [N/A] Kms [N/mm] Separation of the nonlinear distortion components Linear Signal Sound pressure output Music Test signals Loudspeaker Model S LIN P lin S DIS P DIS Distortion Perceptual Signal Quality Analyzer Coil displacement, Power, Temperature Klippel, Tutorial: Loudspeakers at High Amplitudes, 98

Perceptual Sound Quality Evaluation OPERA/PEAQ Total signal Ideal Speaker (Linear) distortion Perceptional attributes Real Speaker Klippel, Tutorial: Loudspeakers at High Amplitudes, 99

How to Specify the Optimal Transducer? Parameters give a comprehensive set of data!! 1. Parameters (independent of stimuli) Acoustical transfer functions Mechanical transfer functions Small signal parameter T/S Large signal parameters (thermal, nonlinear) Should be transformed into parameters 2. Stimulus-based Characteristics Maximal SPL Nonlinear distortion (THD, IMD, XDC) Symptoms of irregular defects (rub, buzz, leakage,...) Coil temperature, compression, Pmax Klippel, Tutorial: Loudspeakers at High Amplitudes, 100

Transformation of Symptoms into Parameters Intermodulation distortion IMD @ SPL MAX force factor BL(x) [Percent] 40 35 30 25 20 15 10 Relative third-order intermodulation distortion ( d3 ) Pfar - pressure in far field 1.00 V 2.15 V 4.64 V 10.00 V Target: KLIPPEL Check 3rd order IMD @ SPL MAX using two-tone signal f 2 >> fs and f 1 < fs Displacement limit X Bl Force factor Bl(X) vs displacement 5 5,0 5 mm 10 mm 0 4*10 2 6*10 2 8*10 2 10 3 2*10 3 Frequency f1 [Hz] 4,5 4,0 Two- tone Stimulus Loudspeaker Model (SIM Module) IMD Distortion Bl [N/A] 3,5 3,0 2,5 2,0 50 % Bl Coíl Height 1,5 change WIDTH in nonlinear curve editor of Bl(x) curve 1,0 0,5 0,0-10,0-7,5-5,0-2,5 0,0 2,5 5,0 7,5 10,0 Displacement X [mm] Klippel, Tutorial: Loudspeakers at High Amplitudes, 101

Design of the Transducer Coupled mechano-acoustical analysis Magnetic FEA FEA BEA u Motor (coil,gap, magnet) F X(r c ) coil former v Mechanical System (suspension, cone, diaphragm) Radiator F(r c ) Acoustical System (enclosure, horn) v(r) near field p(r) Sound Propagation p(r a ) Far-Field T v P Thermal Dynamics Thermal FEA Klippel, Tutorial: Loudspeakers at High Amplitudes, 102

FEA Modeling Progress: Processing time, handling Asymmetries in the shape Geometrical nonlinearities ( variation of the geometry ) Acoustical-mechanical coupling by A. Svobodnik NADwork FineCone By P. Larsen Problems: (Nonlinear) Material parameters visco-elastic properties (Hyperelasticity, creep, Relaxation) Pacsys ANSYS Comsol Klippel, Tutorial: Loudspeakers at High Amplitudes, 103

Equivalent Input Distortion Equivalent Input Fundamental and Distortion in Volt u u D Nonlinear System H(f,r 1 ) H(f,r 2 ) p(r 2 ) H(f,r 3 ) p(r 1 ) sound field p(r 3 ) 9 [V] rms 6 5 4 Compression Fundamental KLIPPEL 3 3 rd -order Harmonic 2 1 2 nd -order Harmonic 0 50 100 200 500 1k Frequency [Hz] Klippel, Tutorial: Loudspeakers at High Amplitudes, 105

Active Speaker Linearization Active Control Loudspeaker z - u H(f,r 1 ) H(f,r 2 ) p(r 1 ) sound field p(r 2 ) Nonlinear System u D u D Nonlinear System H(f,r 3 ) p(r 3 ) Only the equivalent input distortion (EID) can be compensated by an active control system!! Klippel, Tutorial: Loudspeakers at High Amplitudes, 106

New Degrees of freedom Active Speaker System Controller Design Adjustment Passive Speaker Design sound quality (linear and nonlinear distortion) cost,weight directivity enclosure volume max. sound pressure efficiency Klippel, Tutorial: Loudspeakers at High Amplitudes, 107

Force Factor Bl(x) Klippel, Tutorial: Loudspeakers at High Amplitudes, 108

Curing Loudspeaker Defects by DSP? Coil hitting backplate Buzzing loose joint Rubbing voice coil Flow noise at air leak Loose particle hitting membrane vibration Loose particle Deterministic Semi random (mixed characteristic) Random Waveform is completely reproducible Envelope is reproducible (Waveform is not) Waveform is not reproducible Klippel, Tutorial: Loudspeakers at High Amplitudes, 109

Loudspeaker Defect: Voice Coil Rubbing signal contains reproducible and stochastic components Cause: rocking mode at 328 Hz Voice coil gap voice coil rubbing distortion signal one period time Klippel, Tutorial: Loudspeakers at High Amplitudes, 110

Loudspeaker Defect: Air Noise stochastic signal air pressure is changed by coil displacement synchronized with stimulus signal envelope gap cone leakage dust cap Air noise time one period Klippel, Tutorial: Loudspeakers at High Amplitudes, 111

Loudspeaker Defect: Loose Particles cone random process impulsive particles are accelerated by cone displacement not synchronized with stimulus constant output power bouncing gap Voice coil former dust cap Loose Particle bouncing distortion signal one period time Klippel, Tutorial: Loudspeakers at High Amplitudes, 112

Loudspeaker Defect: Voice Coil Bottoming Voice coil Voice coil Voice coil Voice coil backplate backplate backplate backplate distortion signal one period time Short impulse, deterministic symptom Klippel, Tutorial: Loudspeakers at High Amplitudes, 113

Protection of the Driver audio signal Ptrotection System state vector Mirror Filter - parameter vector Adaptive Detector voltage current Benefits: Access to critical state variables (displacement, temperature) automatic adjustment to particular speaker (Xmax, Tv) full mechanical protection due to prediction of envelope minimal impact on sound quality no additional time delay Klippel, Tutorial: Loudspeakers at High Amplitudes, 114

Summary Thermal and nonlinear properties of the loudspeaker limit the maximal acoustical output cause a smooth compression of the fundamental cause additional signal component (distortion) indicate an overload situation can be described by lumped parameters can be predicted by FEM are directly related with cost, weight and size can be be identified by voltage and current monitoring have to be considered by electrical protection systems can be compensated by adaptive control Klippel, Tutorial: Loudspeakers at High Amplitudes, 115

Thank you! Klippel, Tutorial: Loudspeakers at High Amplitudes, 116

Loudspeaker Nonlinearities Causes, Parameters, Symptoms Get a free poster giving a summary on this topic Detailed discussion on practical examples in the Journal of Audio Eng. Soc., Oct. 2006. Poster on cone vibration and sound radation Klippel, Tutorial: Loudspeakers at High Amplitudes, 117