Hot and Nonlinear – Loudspeakers at High Amplitudes - Klippel GmbH

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. 


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 


Amplitude 

X 

[mm] 

30 

10 

3 

1 

0,3 

voice-coil 

displacement 

Assessment of Sound Quality 

Nonlinear 

and 

Thermal 

Model 

Linear 

Model 

Destruction 

Large signal 

performance 

Small signal 

performance 

Related to size, 

weight, cost !!! 

• Maximal Output 

• Distortion 

• Power Handling 

• Stability 

• Compression 

•Bandwidth 

•Sensitivity 

•Flatness of Response 

•Impulse Accuracy 


Watch for Nonlinear Symptoms ! 

Generator 

1. Experiment 

f < f s 

tone at f 

2. Experiment 

f � f s 

Resonance 

frequency f s 

pointer 

scale 

Klippel, Tutorial: Loudspeakers at High Amplitudes, 5 

stroboscope 

3. Experiment 

f > f s

Vibration Behavior 


Klippel GmbH

X [mm] 

7,5 

5,0 

2,5 

0,0 

-2,5 

-5,0 

1. Symptom: Distorted Waveform 

Input: sinousoidal voltage signal at 20 Hz 

Output: Voice coil Displacement 

ZOOM 

Y2(t) 

Input signal Y2(t) vs time 

0,00 0,05 0,10 0,15 

Time [s] 

0,20 0,25 

KLIPPEL 


Distortion in 

the time 

domain

2. Symptom: Harmonic Distortion 

Stimulus Response 

Amplitude 

f1 

frequency 

Amplitude 

DC 


f1 

harmonics 

A single tone generates harmonics and a DC component 

(in displacement) 

Distortion in 

the frequency 

domain 

frequency

3. Symptom: Intermodulation Distortion 

Two-tone Stimulus 

Amplitude 

f1 

bass tone 

harmonics 

sound pressure spectrum 

difference tones summed tones 

f2 

voice tone 

frequency 


dBu (Uo = 1V) 

20 

10 

0 

-10 

-20 

-30 

-40 

-50 

Example: Reproduced Two-tone Signal 

input Nonlinear System 

output 

Response 1 

Frequency Domain 

101 102 103 f [Hz] 

Two-tone signal 

dBu (Uo = 1V) 

20 

10 

0 

-10 

-20 

-30 

-40 

-50 

Response 1 

Frequency Domain 

101 102 103 f [Hz] 

Fundamentals 

Harmonics 

Intermodulation 


X [mm] 

7,5 

5,0 

2,5 

0,0 

-2,5 

-5,0 

ZOOM 

4. Symptom: DC-Displacement 

Y2(t) 

Input signal Y2(t) vs time 

0,00 0,05 0,10 0,15 

Time [s] 

0,20 0,25 

KLIPPEL 

K MS(x) 

•Dc displacement is generated dynamically by rectification of ac components 

•Caused by asymmetrical nonlinearities 

DC displacement 

rest position of the coil 

•working point is shifted away from rest position 

•significant amplitude X DC (comparable with fundamental) 

• usually in displacement (not in velocity, accelaration, input current) 


x

5. Symptom: Amplitude Compression 

2,5 

Linear System 

X [mm] (rms) 

2,0 

1,5 

1,0 

0,5 

0,0 

23.4 Hz 

Fundamental component 

| X ( f1, U1 ) | 

KLIPPEL 

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

Voltage U1 [V] 

10,0 12,5 15,0 


dB - [V] (rms) 

130 

125 

120 

115 

110 

105 

100 

95 

90 

85 

80 

Compression of SPL Output 

for a sinusoidal tone versus frequency 

Sound Pressure Response 

Long Term Response linear response 

Limited by peak displacement 

Limited by coil temperature 

20 50 200 500 2k 

Frequency [Hz] 

KLIPPEL 


Long term response 

was measured by 

using a stepped sine 

wave and cycling 1 

min on/1 min off

Compression of 3 rd -order Harmonic 

Percent 

80 

70 

60 

50 

40 

30 

20 

10 

0 

Third-order harmonic distortion in percent (IEC 60268) 

Signal at IN1 

0.50 V 1.57 V 2.64 V 3.71 V 4.79 V 5.86 V 6.93 V 8.00 V 

Voltage 

4*101 6*101 8*101 102 Frequency f1 [Hz] 

KLIPPEL 


Percent 

25 

20 

15 

10 

5 

Harmonic Distortion 

Second-order harmonic distortion in percent (IEC 60268) 


3.71 V 4.79 V 5.86 V 6.93 V 8.00 V 

4*101 6*101 8*101 102 Frequency f1 [Hz] 

• Nonlinear Distortion depend on frequency and voltage 

• Complicated amplitude characteristic (compression, reduction) 

• Measurement versus amplitude also required (3D measurement) 

KLIPPEL 


| X ( f1, U1 ) | 

Amplitude response of fundamental component in displacement 

10 0 

[mm] 

10 -1 

10 -2 

10 -3 

6. Variation of Resonance Frequency 

caused by suspension nonlinearity K(x) 

6 dB 

steps 

resonance peak 

10 100 200 



7. Symptom: Bifurcation into Multiple States 

X [mm] 

6 

5 

4 

3 

2 

1 

backbone curve 

10 100 

frequency f [Hz] 

Small signal 

resonance 

Sweeping down 

Large signal 

resonance 

Sweeping up 

unstable states 


Occurs if: 

8. Symptom: Instabilities 

- Driver has soft linear suspension 

- Equal-length configuration 

(Bl(x) nonlinearity) 

- Sinusoidal stimulus f > fs 

Paket 

bifurcation caused by motor.MOV 

Bl(x) 

5,0 

4,5 

[N/A] 

4,0 

3,5 

3,0 

2,5 

2,0 

1,5 

1,0 

0,5 

Unstable for f > f s 


KLIPPEL 

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

[mm] x 

� Bifurcation into two stable states of vibration

Bl(x) 

5,0 

4,5 

[N/A] 

4,0 

3,5 

3,0 

2,5 

2,0 

1,5 

1,0 

0,5 

9. Symptom: Chaotic Behavior 

KLIPPEL 

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

[mm] 

x 

� Generation of subharmonics 

x [mm] 

30 

20 

10 

0 

-10 

-20 

-30 


x(t) 

KLIPPEL 

0,0 0,1 0,2 0,3 0,4 0,5 

t [s] 

0,6 0,7 0,8 0,9 1,0 

� Stochastic responses for sinusoidal excitation 

How far away from chaos ?

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, …) 



Maximal Sound Pressure Output 

for a sinusoidal input with different On/Off Cycle 

140 

135 

130 

125 

120 

115 

110 

105 

100 

95 

90 

SOUND PRESSURE LEVEL 

Maximal SPL (short term) 

Cycled sine wave 1 s on, 1 min off 

20 50 100 200 500 1k 2k 5k 

Limited by displacement 

Maximal SPL (long term) 

Cycled sine wave 1 min on, 1 min off 


Limited by coil temperature 

Limited by acceleration 

KLIPPEL 


Agenda 







7. Conclusion 


Loudspeakers - a nonlinear and time-varying system 

Amplitude 

X 

[mm] 

30 

10 

3 

1 

0,3 

voice-coil 

displacement 

Working range 

Destruction 

Large signal domain 

Small signal domain 

MODELING 

Nonlinear Model 

Thermal Model 

Linear Model 


How to Structure the Model ? 

Domains and interfaces passed by the audio signal 

Input power 

Coil temperature 

digital 

Data compression 

Sample rate 

conversion, 

AD conversion 

Voltage 

electrical 

Amplifier 

Crossover 

Motor 

Heat transfer 

Convection cooling 

Time constants 

thermal 

Coil 

displacement 

velocity 

mechanical 

Suspension 

Cone 

Cone 

displacement 

Sound 

pressure in 

ear channel 

acoustical 

Radiation, 

Diffraction 

Propagation, 

Room Influence 


psychoacoustical 

Perception, 

Evaluation

Power density: 

Audio 

signal 

Power Distributed by the Loudspeaker 

Amplifier 

Crossover 

EQ 

linear 

u(t) 

10 W / cm 3 1 W / cm 3 1 W / m 3 1 mW / m 3 

Electromechanical 

Transducer 

i(t) 

x(t) 

nonlinear 

Mechanoacoustical 

Transducer 

(Cone) 

Radiation 



Sound 

Propagation 

Sound 


Sound 


linear 

Room 

Acoustics 

Room 

Interference 

Room 

Interference 


p(r 1 ) 

p(r 2 ) 

sound 

field 

p(r 3 )

estoring 

force 

Stiffness K ms (x) of Suspension 

F � Kms 

( x) 

x 

displacement 

F 

F 

x 

x 

K 

6 

N/mm 

5 

4 

3 

2 

1 

spider 

-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 

Kms(x) determined by 

• suspension geometry 

• impregnation 

surround 

diplacement x mm 

• adjustment of spider and surround 


total 

suspension

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Mechanical Stress in a Spider 


Symmetrical Limiting of Spider 

K_MS [N/mm] 

4,0 

3,5 

3,0 

2,5 

2,0 

1,5 

1,0 

0,5 

0,0 

-x_max < x < x_max 

stiffness K_MS(x) 

KLIPPEL 

-10 -5 0 5 10 

> 

spider 

diaphragm 

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 


frame

Distortion Generated by K ms(x) 

simplified signal flow chart 

fs 

Voltage pressure 

distortion 

multiplier 

Displacement x 

� ( x) 

� K ( 0) 

� 

Kms ms Re 

� 

Bl 

fs 

lowpass 

highpass 

Bass tone 

� Multiplication of displacement time signals x(t)*Kms(x(t)) 


pass band

Symptom of Kms: Amplitude Compression 

Fundamental coil displacement in a vented-box System 

linear 

nonlinear 

X [mm] (rms) 

6 

5 

4 

3 

2 

1 

0 


| X ( f1, U1 ) | 

1.00 V 2.00 V 3.00 V 

4.00 V LINEAR MODEL 

Compression 

101 102 Frequency f1 [Hz] 


Nonlinear Stiffness K ms(x) 

Symptoms: 

- Compression of the fundamental (f

ack plate 

F 

magnet 

Φ dc 

Force Factor Bl(x) 

pole plate 

B-field 

0 mm 

F � Bl( 

x) 

i 

U � Bl( 

x) 

v 

coil 

pole piece 

displacement 

Electro-dynamical driving force Voice coil current 

Back EMF Voice coil velocity 

x 

Bl(x) determined by 

• Magnetic field distribution 

• Height and overhang of the coil 

• Optimal voice coil position 

3.5 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

0.0 


5.0 

N/A 

Bl(x) 

N S 

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 

> 

F 

i 

F � 

Bli

Motor with Equal-length Configuration 

magnet 

pole piece 

h coil 

h 

gap 

Coil height h coil � gap height h gap 

x 


voice coil 

Properties: 

3,0 

50% 

b [N/A] 


3,5 

2,5 

2,0 

1,5 

1,0 

0,5 

0,0 


force factor b(x) 

h coil 

KLIPPEL 

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

> 

• Sensitive to offset in rest position 

• Sensitive to instabilities f>fs • Low-order distortion at low amplitudes 

• Low inductance and flux modulation 

�

magnet 

pole piece 

Motor with Overhang Coil 

h coil 

h 

gap 

Coil height h coil > gap height h gap 

x 


voice coil 

Properties: 

b [N/A] 

5,5 

5,0 

4,5 

50% 

4,0 

3,5 

3,0 

2,5 

2,0 

1,5 

1,0 

0,5 

0,0 


force factor b(x) 

�h � h � 

h coil 


� 

� 

coil 

KLIPPEL 

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

> 

• 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 

gap

Adjusting Coil‘s Rest Position 

Bl [N/A] 

5,0 

4,5 

4,0 

3,5 

3,0 

2,5 

2,0 

1,5 

1,0 

0,5 

0,0 

magnet 

pole piece 

x=0 


Induction B 

voice coil 

displacement 

Force factor Bl vs. displacement X 

Bl(X) 

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

Displacement X [mm] 

Bl [N/A] 

5,0 

4,5 

4,0 

3,5 

3,0 

2,5 

2,0 

1,5 

1,0 

0,5 

0,0 

magnet 

pole piece 

x=x b 


Induction B 

voice coil 

Force factor Bl vs. displacement X 

Bl(X) 

displacement 

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



Generation of Bl(x) Distortion 

1 st nonlinear effect: Parametrical Excitation 

Voice tone 

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) 

fs 

fs 


pass band 

Voltage fs pressure 

distortion 

impedance highpass 

multiplier 

current 

x 

lowpass

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) 

Amplitude Modulation 

KLIPPEL 

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

[mm] 

two-tone stimulus f 1 < f s, f 2 > f s 

Pfar [ N / m^2 ] 

5,0 

2,5 

0,0 

-2,5 

-5,0 

Sound pressure Pfar(t) in far field vs time 

Pfar(t) 

x peak 

Bottom 

Rest position 

Mean 

Cycle 

0,05 0,10 0,15 

Time [s] 

0,20 0,25 0,30 


Third-order intermodulation distortion in percent (IEC 60268) 

Pressure Pfar in far field 

125 

100 

75 

50 

25 

Symptoms of Bl(x) 

Intermodulation Distortion 

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 

4*102 6*102 8*102 103 Frequency [Hz] 

KLIPPEL 

(constant bass tone f 1 =0.5f s and varying voice tone). 

magnet 

pole piece 

x=x b 



Induction B 

voice coil 

displacement 

Parametric excitation 

F=Bl(x)*i

X dc 

5 mm - 

0 

Bl-maximum 

dc-displacement (X dc ) 

Symptoms of Bl(x) 

dc displacement 

unstable 

fs 2fs frequency 

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) 

UNIQUE SYMPTOM 


(x) 

Instability of the Nonlinear Motor 

5,0 

4,5 

[N/A] 

4,0 

3,5 

3,0 

2,5 

2,0 

1,5 

1,0 

0,5 

Unstable for f > f s 

KLIPPEL 

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

[mm] x 

145.0 

U [V] 

20 

15 

Effects: 

• Reduced acoustical output 

• Substantial Distortion 


10 

5 

Fundamental 

| Pfar( f1, U1 ) | 

100 

200 

300 

f [Hz] 

400 

0.75 

0.50 

0.25 

0.00 

500 

Pfar [N/m^2]

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 

0,0 

Curve 1 

Effect of a Nonlinear Motor 

KLIPPEL 

-15 -10 -5 0 5 10 15 


• Parametric excitation 

F=Bl(x)*i 

• Nonlinear damping 

F=Bl(x) 2 /Re*v 

mm 

8 

7 

6 

5 

4 

3 

2 

1 

0 

driver with 

• dominant Bl(x)-nonlinearity 

• linear suspension 

• constant inductance 


| X ( f1, U1 ) | 

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 

102 103 Frequency [Hz] 



Symptoms: 

- Compression of the fundamental (f

Φ coil(-9 mm) 

U ind 

Φ counter 

Reluctance 

force 

shorting ring 

d ( x, 

i) 

� � 

dt 

� 

F rel 

Voice Coil Inductance L e(x) 

-9 mm 

d 

�L( x) 

i� 

dt 

0 mm 

2 

i ( t) 

dL( 

x) 

� � 

2 dx 

Φ coil(+9 mm) 

9 mm 

Differentiated 

Magnetic flux 

voice coil displacement 

x 

L e(x) determined by 

With with shorting rings ring 

• geometry of coil, gap, magnet 

• optimal size and position of short cut ring 


4.0 

Le 

[mH] 

2.5 

2.0 

1.5 

1.0 

0.5 

0.0 

Without without shorting rings ring 

-15 -10 -5 0 5 10 15 

[mH 

0,30 

0,25 

0,20 

0,15 

0,10 

0,05 

0,00 

Effect of Nonlinear Inductance L e(x) 

Le(X) 

[Ohm] 

25 

20 

15 

10 

5 

0 

KLIPPEL 

-4 -3 -2 -1 -0 1 2 3 4 

> 

X=-4 mm X=4 mm 

Le(x) nonlinearity causes variation of 

electrical input impedance 

Magnitude of electric impedance Z(f) 

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

Coil is clamped 

X=-4 mm 


KLIPPEL 

X=4 mm 

101 102 103 104 Frequency [Hz]

Distortion caused by L e(x) 

6dB/oct 

pass band 

fs 

Voltage pressure 

distortion 

differentiator 

multiplier 

L( 

x) 

1� L(x) L( 

0) 

current 

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) 

i 

x 

displacement 

impedance 

fs 

fs 

lowpass 

highpass 


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 


Counter flux 

� C 

L E(x) 

L_E [mH] 

Remedy for L e(x): Shorting Ring 

1,75 

1,50 

1,25 

1,00 

0,75 

0,50 

0,25 

0,00 

magnet 

pole piece 


inductance L_E(x) 


Coil flux � A 

voice coil 

KLIPPEL 

Reduced 

value 

Less variation versus x 

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

> 

short cut ring 

x 

i 

R E (T V ) 

u 

L E (x) 

R RING 

Effect depends on 

b(x)v b(x) b(x)i 


L 2 (x) 

R 2 (x) 

v 

CMS (x) MMS RMS Electro-mechanical Equivalent Circuit 

• Geometry (Ring or Cap) 

• Material (Aluminum or Copper) 

• Size, position, distance to the coil 

F m (x,i)

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 


• V < V Crit 

Flow Resistance R A(v) of a Port 

at Medium Amplitudes 

• Generating an air jet 

• energy dissipated in the far field 

• Harmonics at low frequencies 

R A(v) 

• R A(v) ~ |v|*m v 


Asymmetrical Flow Resistance R A(v) 

p box 

• R A(v 0) > R A(-v 0) 

p 0 

x DC 

• DC component in pressure p box 

• dynamical voice coil offset x 

R A(v) 

p box 


v 

x DC

Symptoms: 

Vented Box System 

• 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 !! 


Nonlinear Sound Propagation 

Wave Steepening 

Speed of sound depends on sound pressure c(p) 

Sound pressure maxima travel faster than minima 


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 


Nonlinear Mechanical Resistance R ms(v) 

Air flow 

magnet 

Pole piece 

v 

dome 

gap 

spider 

R ms(v) 

R ms(v) depends on velocity v of the coil due to air flow and turbulences at vents and 

porous material (spider, diaphragm) 


v

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 


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 


Ranking List of 

Transducer Nonlinearities 

1. Force Factor Bl(x) 

2. Compliance Cms(x) � tweeter 

3. Inductance Le(x) 4. Flux Modulation of Le(i) 5. Mechanical Resistance Rms(v) 6. Nonlinear Sound Propagation c(p) 

7. Doppler Distortion �(x) 

8. Flux Modulation of Bl(i) 

9. Nonlinear Cone Vibration 

10. Port Nonlinearity RA(v) 11. many others ... 

� microspeaker 


� woofers 

� microspeaker 

� horns

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 

i 

Re (Tv) 

Le(x) 

L2(x) 

R2(x) 

Mms Rms(v) Cms(x) 

U Bl(x,I)V Bl(x,I) Bl(x,I)I 

Sd 

V 

Fm (x,I) 

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 

Cab(pbox) 

Cr(prear) 


SdV 

pbox 

prear 

Ral 

qp 

Rap(qp) 

Map 

radiation

Signal Flow Chart of the Transducer 

simplified by assuming a linear impedance for the mechanical load 

Feedback 

Nonlinear differential equation 

Based on lumped parameter modeling 

Linear system modeled by transfer function H(f,r a) 

Feed-forward 


Source of the Nonlinear Distortion 

considering dominant nonlinearities in electrodynamical loudspeakers 

Loudspeaker 

u 

input 

uD 


distortion 


System 

H(s,r1) 

H(s,ri) p(ri) 

H(s,rN) 

p(r1) 

sound 

... 

... field 

... 

... 

... 

... 

p(rN) 


Voice Coil Heating depends on the 

Spectral Properties of the Stimulus 

100 

90 

80 

70 

50 

30 

20 

10 

0 

-1 0 

KLIPPEL 

[K ] [W ] 

40 

t 1 

t 2 

0 250 500 750 1000 1250 1500 1750 2000 2250 

t [s e c ] 

t 3 

P RE 

P Re 

� T V 

Delta Tv 

Music: Classic 

Pop 

Vocal 

Thermal resistance 

�T v/P re= 

6,8 K/W 4,6 K/W 7,5 K/W 

High voice coil displacement gives 

high Convection cooling 


30 

25 

15 

10 

is not constant !! 

5 

0

P coil 

T v 

�T v 

Nonlinear Thermal Model 

C tv 

P tv 

P con 

R tc (v) 

Air 

convection 

cooling 

Rta (x) 

R tv 

R tt (v) 

C ta 

T a 

P eg 

Direct heat 

transfer 

T g 

�T g 

�T m 


P g 

C tg 

R tg 

P mag 

T m 

C tm 

R tm

Agenda 







7. Conclusion 


Identification of the Driver Model 

Three Tasks: 

Abstraction 

i 

R E (T V ) 

u 

L E (x) 

L 2 (x) 

R 2 (x) 

b(x)v b(x) b(x)i 

Lumped Parameter Model 

Identification 

v 

CMS (x) MMS RMS • 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, ...) 


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 

Differential Equation

Large Signal Model & Parameters 

Heat transfer 

Convection cooling 

Stimulus 

Amplifier 

Crossover 

Linear 

System 

H el(s) 

R tv, R tm, 

� tv, � tm, 

Bl(x), L(x), 

L(i) 

T/S Parameter 

Thermal 

Model 


System 

Kms(x) 

Z m(s) 

Motor Suspension Cone 

Linear 

System 

H a(s) 


Sound 

pressure 

output 

Enclosure, Horn room

Measurement of Large Signal Parameters 

Standard IEC 62458 

defines 

1. Static (quasi-static) method 

2. Incremental dynamic method 

3. Full dynamic method 


Method: 

1. Select a working point 

2. Excite with DC stimulus 

Static Measurement 

3. Measure associated state signals 

4. Calculate instantaneous parameters 

5. Repeat the measurement at other 

working points 


F 

secant 

x 

K MS(x)= F DC 

x DC

Example: Static Measurement of Stiffness 

Ringlstetter Harman Becker Straubing 2004 


Incremental dynamic Measurement 

Method: 

1. Select a working point 

2. Excite with a dc stimulus 

3. Add a small AC-signal 

4. Measure state variables XAC and FAC 5. Calculate gradient Kgrad (XDC) 6. Repeat at other working points 

7. Parameter Transformation 

F DC FAC 

K grad(x)= F AC 

X AC 

Transformation 

K MS(x)= F 

X 


tangent 

x DC 

x AC

First available product: DUMAX 

incremental dynamic method 

Photo courtesy by D. Clark 


Method: 

Full dynamic Measurement 

1. Excite speaker with audio-like signal 

2. Measure instantaneous state variables 

3. Estimate free model parameters 

� first application of system 

identification to loudspeakers 

(Knudsen 1993) 

F AC(t) 

K MS(x)= F 

X 


F 

secant 

x 

X AC(t)

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 


Adaptive Identification Principle 

full dynamic method based on current & voltage measurement 

audio-like, 

persistent 

Signal 

Source 

protection 

Detection 

working range 

normal 

working 

conditions 

Model 

precise, 

robust 

sensor 

State 

Measurement 

optimal fitting 

Parameter 

Estimation 


Stimulus 

Noise, 

Audio signals 

(music, noise) 

Multi-tone 

complex 

Linear Parameters 

• T/S parameters at x=0 

• Box parameters fb,Qb 

• Impedance at x=0 

Dynamic Measurement 

of Motor and Suspension Nonlinearities 

Voltage & current 


System 

Identification 

Nonlinear Parameters 

• nonlinearities Bl(x), Kms(x), Cms(x), 

Rms(v), L(x), L(i) 

• Voice coil offset 

• Suspension asymmetry 

• Maximal peak displacement (Xmax) 


State Variables 

• peak displacement during measurement 

• voice coil temperature 

• eletrical input power, 

Thermal Parameters 

• Thermal resistances Rtv, Rtm 

• Thermal capacity Ctv, Ctm 

• Air convection cooling

Controlling the Production Process 

Corrected 

Rest Position 

B field 

voice 

coil 

Shift coil 0.6 mm to backplate 

Action 

Failure: 

Coil Offset 

B field 

Fail 

voice 

coil 

Targets: 

•Detection of motor and 

suspension problem as fast 

as possible (when first 

device arrives at the end of 

line) 

•Simplify interpretation of 

results (voice coil offset in 

mm) 

•Single-valued parameter 

can be directly be used for 

adjustment 


Delta Tv [K] 

[mm] 

20 

10 

0 

-10 

-20 

125 

100 

75 

50 

25 

0 

Measurement of Thermal Parameters 

within the Large Identification Module (LSI) 

0 Delta 2500 Tv 5000 7500 P real 10000 12500P Re 15000 

t [sec] 

KLIPPEL 

nonlinear 

r V 

0 2500 5000 7500 

t [sec] 

10000 12500 15000 

R TV ,� V 

� 

Xpeak Xbottom 

Voice coil displacement 

Power dissepated in Re 

R TM ,� M 

Thermal identification 


KLIPPEL 

Voice coil temperature 

Real input power 

225 

200 

175 

150 

125 

100 

75 

50 

25 

0 

P [W]

Agenda 







7. Conclusion 


Small Signal Performance 

Specifications for Active and Passive Loudspeaker Systems 

180° 

90° 

90° 

270° 

-90° 

� 

� 

0° 

All information are provided by a complex 3D far 

field response H(f,�, �) measured with sufficient 

angular resolution 

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 


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 


How to interprete Harmonic Distortion 

2 nd harmonic 

asymmetry 

2 nd , 3 rd and higher-order harmonics 

(amplitude and phase information) 

3 rd harmonic 

symmetry 

THD 

magnitude 

Characteristics of the nonlinearity 


Important for 

irregular defects 

„rub & buzz“ 

CHD 

Crest factor 

smoothness


130 

120 

110 

100 

90 

80 

70 

60 

50 

Interpretation of THD in SPL 

Kms(x) 

Bl(x) 

L(x) 

Fundamental 

Fundamental 

Fundamental THD 

50 100 200 500 1k 2k 5k 


THD 

L(i) 

Cone Vibration 


KLIPPEL

dB - [V] 

0 

-5 

-10 

-15 

-20 

-25 

-30 

-35 

-40 

-45 

Transform Distortion to the Source 

Sinusoidal 

sweep 

Distortion in Voltage 

3rd harmonic distortion in voltage 


nearfield 30 cm 60 cm 1 m distance 

KLIPPEL 

Independent of room 

50 100 200 


500 1k 

U(f) 

D 



System 

System 

H(f,r1) 

H(f,r2) 

Inverse filtering 

with H(f,r) 

p(r1) 

sound 

field 

p(r2) 

3rd harmonics absolute 


1 m distance 60 cm distance 30 cm distance nearfield 

KLIPPEL 


dB - [V] 

90 

85 

80 

75 

70 

65 

60 

55 

50 

45 

40 

Sound pressure 

measurement 

50 100 200 


500 1k

Interpretation of Multi-tone Distortion 

[dB] 

120 

110 

100 

90 

80 

70 

60 

50 

Kms(x) 

Fundamental Multi-tone Distortion 

Fundamental 

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


Bl(x) 

Distortion 

L(x) 

L(i) 

Doppler Effect 

Cone Vibration 

KLIPPEL 


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 


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) 


P coil 

P RE + 

0.3 (P real –P RE) 

T v 

�T v 

C tv 

Thermal Analysis 

single tone at 65 Hz with 40 V rms 

P con 

Air 

convection 

cooling 

P tv 

R tc (v) 

1/r tv V rms 

R tv 

0.93 K/ W 


transfer 

T a 

P eg 

�T m 

C tm 


P g 

0.7 (P real –P RE) 

T m 

0.37 K/ W 

R tm

P coil 

167 W 

T v 

�T v 

C tv 



146 W 

Rtv 196 K P T g 163 W m 60 K 

P tv 

P con 

Air 

convection 

cooling 

21 W 

R tc (v) 


transfer 

T a 

P eg 

17 W 

�T m 

C tm 


R tm

70 W 

P coil 

T v 

�T v 

C tv 



69 W 

P tv 

P con 

R tc (v) 

R tv 

99 K 34 K 

1 W 

T a 

P eg 

22 W 


transfer 

�T m 

C tm 


P g 

91 W 

T m 

R tm

P coil 

134 W 

T v 

�T v 


two-tone at 1000 Hz and 50 Hz with 30 V rms 

C tv 

118 W 

R tv 

160 K 50 K 

P tv 

P con 

16 W 

R tc (v) 


transfer 

T a 

P eg 

16 W 

�T m 

C tm 


P g 

134 W 

T m 

R tm

P coil 

Tv Rtv Tm 

�T v 

C tv 

Power Bypass Factor 

The power P tv should be minimal ! 

P tv 

P con 

R tc (v) 

T a 

118 W 

16 W 

16 W 

P eg 

P mag 

�T m 

C tm 

R tm 

� � 


The bypass 

factor � should 

be maximal ! 

P � P 

con eg 

P � P 

Re R2 

Total input power

Optimal Thermal Design 

Investigation of Design Choices 

dome 

vent 

gap 

Vented pole piece 

Av 

dome 


vent 

Av 

gap 

Sealed pole piece 

Which design Choice provides a better heat transfer ?

Bypass Power Factor � 

dome 

vent 

Av 

(vented and sealed pole piece) 

gap 

60 

50 

�� 

40 

[%] 

30 

20 

10 

00 

1 10 100 1000 1000 

frequency [Hz] 

Closing the vent in the pole piece: 

50 percent of the power will bypass the coil ! 

vent sealed 

vent open 


Agenda 







7. Conclusion 


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 


Modeling 

Measurement 

Application 

Parts 

(cone, spider, 

motor parts) 

Loudspeaker Development 

Distributed Parameter 

Models 

(FEA, BEA) 

Geometry 

Material 

Parameter 

Driver 

(woofer, 

tweeter) 

Thermal 

Parameters 

Linear T/S 

Parameters 

Lumped 

Parameter 

Models 

Cone 

Vibration 

Parameters independent of stimulus 


Parameters 

Definition of Target Performance 

Selection of Components & Design 

System 

(driver + Xover + room) 

Evaluation of the First Prototype 

End-of-line Testing 

System-oriented 

Models 


Distortion 

Power 

Handling 

Loudness 

density 

Small Signal 

Performance 


Listener 

Psychoacoustical 

Models 

Symptoms dependent on stimulus 

R&D 

Sound 

Attributes 

Quality 

Metrics 

QC

How to define the Target Performance 

Auralization using the large signal model 

Development 

Manufacturing 

Objective Evaluation 

• Distortion, Maximal Output 

• Displacement, Temperature 

• Evaluation of Design Choices 

• Indications for Improvements 

Marketing 

Management 

Subjective Evaluation 

• Personal Impression 

• Sufficient Sound Quality 

• Tuning to the target market 

• Performance/Cost Ratio 


Auralization - Systematic Listening Tests 

using Simulation and Decomposition Techniques 

Stimulus 

H el(s) 

Linear 

System 

Transfer function 

Thermal 

Parameters 

R tv, R tm 

Power 

varied parameter 

Thermal 

Model 

Lumped 

Parameters 


System 

Bl(x), L(x). 

Cms(x) Re, Mms 

Temperature 

Y(s) 

Mechancial 

Cone 

Admittance 

Linear 

System 

H a(s) 

Electrical Transfer 

Function 

Amplifier Crossover Motor Suspension Cone Enclosure, Horn Room 


Listening test 

Sensations 

Psychoacoustical 

Model

Music 

Test signals 

Nonlinear Auralization Technique 

Force factor Bl (X) 

-Xprot < X < Xprot Xp- < X < Xp+ 

Stiffness of suspension Kms (X) 

6 

KLIPPEL 

5 

4 

3 

2 

2,25 

-Xprot < X < Xprot 

2,00 

1,75 

Xp- < X < Xp+ 

KLIPPEL 

1 

1,50 

0 

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

X [mm] 

1,25 

1,00 

0,75 

0,50 

0,25 

0,00 

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

X [mm] 

5,0 7,5 

Bl [N/A] 

Parameters 

Kms [N/mm] 

Loudspeaker 

Model 

Coil displacement, 

Power, Temperature 

Linear Signal 

SLIN 

SDIS 

Plin 

PDIS 

Distortion 

Separation of the nonlinear 

distortion components 

Perceptual 

Signal Quality 

Analyzer 


Sound pressure 

output

Ideal Speaker (Linear) 

Real Speaker 

Perceptual Sound Quality Evaluation 

OPERA/PEAQ 

Total signal 

distortion 

Perceptional attributes 


How to Specify the Optimal Transducer ? 

Parameters give a 

comprehensive 

set of data !! 

Should be 

transformed into 

parameters 

1. Parameters (independent of stimuli) 

• Acoustical transfer functions 

• Mechanical transfer functions 

• Small signal parameter T/S 

• Large signal parameters (thermal, nonlinear) 

2. Stimulus-based Characteristics 

• Maximal SPL 

• Nonlinear distortion (THD, IMD, XDC) 

• Symptoms of irregular defects (rub, buzz, leakage,...) 

• Coil temperature, compression, Pmax 


Transformation of Symptoms into Parameters 

Intermodulation distortion IMD @ SPL MAX � force factor BL(x) 

[Percent] 

40 

35 

30 

25 

20 

15 

10 

5 

0 

Two- tone 

Stimulus 

Relative third-order intermodulation distortion ( d3 ) 

Pfar - pressure in far field 

1.00 V 2.15 V 4.64 V 10.00 V 

KLIPPEL 

4*102 6*102 8*102 103 2*103 Frequency f1 [Hz] 

Loudspeaker 

Model 

(SIM Module) 

Target: 

change WIDTH in nonlinear 

curve editor of Bl(x) curve 

IMD 

Distortion 

• Check 

• 3rd order IMD @ SPLMAX using two-tone 

signal f2 >> fs and f1 < fs 

• Displacement limit XBl Bl [N/A] 

5,0 

4,5 

4,0 

3,5 

3,0 

2,5 

2,0 

1,5 

1,0 

0,5 

Force factor Bl(X) vs displacement 

50 % Bl 

5 mm 10 mm 

Coíl Height 

0,0 

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



u 

Tv 

Magnetic 

FEA 

Motor 

(coil,gap, 

magnet) 

Thermal 

Dynamics 

Thermal 

FEA 

Design of the Transducer 

P 

F 

coil 

former 

v 

Mechanical 

System 

(suspension, 

cone, 

diaphragm) 

Coupled mechano-acoustical analysis 

FEA 

X(rc) 

Radiator 

Acoustical 

System 

(enclosure, 

horn) 

near field 

F(rc) 

v(r) 

p(r) 

BEA 

Sound 



p(ra) 

Far-Field

Progress: 

• Processing time, handling 

• Asymmetries in the shape 

• Geometrical nonlinearities 

(“variation of the geometry”) 

• Acoustical-mechanical coupling 

Problems: 

• (Nonlinear) Material parameters 

• visco-elastic properties 

(Hyperelasticity, creep, 

Relaxation) 

FEA Modeling 

by A. Svobodnik 

NADwork 

Pacsys 

ANSYS 


FineCone 

By P. Larsen 

Comsol

Agenda 







7. Conclusion 


u 

u D 

Equivalent Input Distortion 


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 

3 

2 

1 

0 

Equivalent Input Fundamental and 

Distortion in Volt 

Compression 

3 rd -order Harmonic 

2 nd -order Harmonic 

Fundamental 


KLIPPEL 

50 100 200 500 1k 

Frequency [Hz]

Active Speaker Linearization 

Active Control Loudspeaker 

z 


System 

- 

u D 

u 

u D 


System 

H(f,r 1 ) 

H(f,r 2 ) 

H(f,r 3 ) 

p(r1 ) 

sound 

field 

Only the equivalent input distortion (EID) can be 

compensated by an active control system !! 

p(r 2 ) 

p(r 3 ) 


New Degrees of freedom 

Controller Design Adjustment 

sound quality 

(linear and nonlinear 

distortion) 

Active Speaker System 

Passive Speaker 

Design 

cost,weight 

directivity 

enclosure volume 

max. sound pressure 

efficiency 




Curing Loudspeaker Defects by DSP ? 

Coil hitting 

backplate 

Waveform is 

completely 

reproducible 

Buzzing loose 

joint 

vibration 

Rubbing 

voice coil 

Envelope is 

reproducible 

(Waveform is not) 

Flow noise at air 

leak 

Waveform is not 

reproducible 


Loose particle 

hitting 

membrane 

Semi‐random 

Deterministic Random 

(mixed characteristic) 

Loose particle

gap 

Loudspeaker Defect: Voice Coil Rubbing 

• signal contains reproducible and 

stochastic components 

Voice coil 

voice coil rubbing 

distortion signal 

Cause: rocking mode at 328 Hz 

one period 


time

Loudspeaker Defect: Air Noise 

• stochastic signal 

• air pressure is changed by coil displacement 

• synchronized with stimulus – signal envelope 

one period 

Air noise 

gap 


cone 

leakage 

dust cap 

time

Loudspeaker Defect: Loose Particles 

• random process 

• impulsive 

• particles are accelerated by cone displacement 

• not synchronized with stimulus 

• constant output power 

bouncing 


bouncing 

one period 

gap 

Voice 

coil 

former 


cone 

dust cap 

Loose Particle 

time

Loudspeaker Defect: Voice Coil Bottoming 

Voice coil Voice coil 

Voice coil 

backplate backplate backplate backplate 


Short impulse, deterministic symptom 

one period 

Voice coil 


time

Protection of the Driver 

audio 

signal 

Ptrotection 

System 

parameter 

vector 

state 

vector 

Mirror Filter 

- 

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 


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 


Thank you ! 


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

Hot and Nonlinear – Loudspeakers at High Amplitudes - Klippel GmbH

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