KLIPPEL ANALYZER SYSTEM Large Signal Identification (LSI) Including Thermal Data

 Driver Name: SS 9 Driver Comment: Measurement: LSI Woofer Driver (Thermal) Measurement Comment: Date: 07/21/07 Time: 19:39:03

Obvious shorting method

BL and KMS symmetry is almost perfect

Note XBl and XC are calculated using the subwoofer distortion targets of 10% as this driver is to be used as a midbass (Using 10% values), the XBl is 13.4mm and the XC is 10.6mm

# Overview

This Report illustrates the powerful features of the Large Signal Identification Module (LSI):

< nonlinear speaker parameters versus displacement and current

< coefficients of the power series expansion of the nonlinear parameters

< derived speaker parameters such as resonance frequency and loss factors

< parameters at the rest position (parameters for linear modelling)

< parameter variation versus time

< state variables of the speaker (temperature, displacement, …)

< contribution of each nonlinearity to the total distortion (distortion analysis)

< suggestions for loudspeaker improvements (remedy parameters)

# Nonlinear Characteristics

The dominant nonlinearities are modelled by variable parameters such as

 Bl(x) instantaneous electro-dynamic coupling factor (force factor of the motor) defined by the integral of the magnetic flux density B over voice coil length l as a function of displacement KMS(x) mechanical stiffness of driver suspension a function of displacement LE(i) voice coil inductance as a function of input current (describes nonlinear permeability of the iron path) LE(x) voice coil inductance as a function of displacement

The electrodynamic coupling factor, also called Bl-product or force factor Bl(x), is defined by the integral of the magnetic flux density B over voice coil length l, and translates current into force.

In traditional modeling this parameter is assumed to be constant. The force factor Bl(0) at the rest position corresponds with the Bl-product used in linear modeling.

The red curve displays Bl over the entire displacement range covered during the measurement. You see the typical decay of Bl when the voice coil moves out of the gap.

At the end of the measurement, the black curve shows the confidential range (interval where the voice coil displacement in this range occurred 99% of the measurement time). During the measurement, the black curve shows the current working range.

The dashed curve displays Bl(x) mirrored at the rest position of the voice coil – this way, asymmetries can be quickly identified.

Since a laser was connected during the measurement, a "coil in / coil out" marker is displayed on the bottom left / bottom right.

More information regarding Bl(x) and its optimization can be found in the article Optimal Voice Coil Rest Position

The stiffness KMS(x) describes the mechanical properties of the suspension. It's inverse is the compliance CMS(x)

The inductance components Le (x) and Bl(i)  of most drivers have a strong asymmetric characteristic. If the voice coil moves towards the back plate the inductance usually increases since the magnetic field generated by the current in the voice coil has a lower magnetic resistance due to the shorter air path.

The nonlinear inductance Le(x) has two nonlinear effects. First the variation of the electrical impedance with voice coil displacement x affects the input current of the driver. Here the nonlinear source of distortion is the multiplication of displacement and current. The second effect is the generation of a reluctance force which may be interpreted as an electromagnetic motor force proportional to the squared input current.

The flux modulation Bl(i) has two effects too. On the electrical side the back EMF Bl(i)*v produces nonlinear distortion due to the multiplication of current i and velocity v. On the mechanical side the driving force F = Bl(i)*i comprises a nonlinear term due to the squared current i. This force produces similar effects as the variable term Le(x) in the

# Nonlinear and  thermal Parameters

The displacement limits XBL, XC, XL and Xd describe the limiting effect for the  force factor Bl(x), compliance Cms(x), inductance Le(x) and Doppler effect, respectively, according to the threshold values Blmin, Cmin, max and d2 used. The thresholds Blmin= 82 %, Cmin=75 %, max=10 % and d2=10% generate for a two-tone-signal (f1=fs, f2=8.5fs) 10 % total harmonic distortion and 10 % intermodulation distortion. The thresholds Blmin= 70 %, Cmin=50 %, max=17 % create 20 % total harmonic distortion which is becoming the standard for acceptable subwoofer distortion thresholds.

The thermal parameters describe the resistance and capacity of the coil and magnet and the velocity depending convection cooling.

This new model for thermal behavior of a speaker is detailed in the paper: Nonlinear Modeling of Heat Transfer.

 Symbol Number Unit Comment Displacement Limits thresholds can be changed in Processing property page X Bl @ Bl min=70% 13.5 mm Displacement limit due to force factor variation X C @ C min=50% >14.5 mm Displacement limit due to compliance variation X L @ Z max=10 % 5.0 mm Displacement limit due to inductance variation X d @ d2=10% 48.0 mm Displacement limit due to IM distortion (Doppler) alpha 0.259304 Heating of voice coil by eddy currents alphaOrg Heating of voice coil by eddy currents (without limits) Rtv 1.207180 K/W thermal resistance coil ==> pole tips rv 0.666803 Ws/Km air convection cooling depending on velocity Rtm 0.550532 K/W thermal resistance magnet ==> environment tau m 40 min thermal time constamt of magnet Ctm 4385.657715 Ws/K thermal capacity of the magnet tau v 115.219452 s thermal time constant of voice coil Ctv 95.445122 Ws/K thermal capacity of the voice coil delta Tw 95 K Temperature increase in Warm Resistance Mode delta Tc 94 K Temperature increase in Convection Mode delta Te 38 K Temperature increase in Eddy Mode Pcoil(warm) 83.189354 W Pcoil in warm mode Pcoil(conv) 109.966309 W Pcoil in convection mode Ptv(mag.beg) 48.910084 W power heating the coil at beginning of magnet mode Ptv(mag.mid) 49.003696 W power heating the coil sampled in the middle of magnet mode Ptv(mag.end) 48.442669 W power heating the coil at end of magnet mode Ptm(mag.beg) 99.213829 W power heating the magnet at beginning of magnet mode Ptm(mag.mid) 98.905685 W power heating the magnet sampled in the middle of magnet mode Ptm(mag.end) 97.308136 W power heating the magnet at end of magnet mode f1 -0.004977 1/A coefficient (1) of Inductance over current (flux modulation) f2 -0.000121 1/A^2 coefficient (2) of Inductance over current (flux modulation) Bl0 = Bl (X=0) 26.330 N/A constant part in force factor Bl1 0.11557 N/Amm 1st order coefficient in force factor expansion Bl2 -0.035345 N/Amm^2 2nd order coefficient in force factor expansion Bl3 -8.5741e-006 N/Amm^3 3rd order coefficient in force factor expansion Bl4 3.2780e-006 N/Amm^4 4th order coefficient in force factor expansion Bl5 N/Amm^5 5th order coefficient in force factor expansion Bl6 N/Amm^6 6th order coefficient in force factor expansion Bl7 N/Amm^7 7th order coefficient in force factor expansion Bl8 N/Amm^8 8th order coefficient in force factor expansion L0 = Le (X=0) 4.0482 mH constant part in inductance L1 -0.088895 mH/mm 1st order coefficient in inductance expansion L2 0.0024967 mH/mm^2 2nd order coefficient in inductance expansion L3 8.0338e-005 mH/mm^3 3rd order coefficient in inductance expansion L4 -1.3017e-006 mH/mm^4 4th order coefficient in inductance expansion L5 mH/mm^5 5th order coefficient in inductance expansion L6 mH/mm^6 6th order coefficient in inductance expansion L7 mH/mm^7 7th order coefficient in inductance expansion L8 mH/mm^8 8th order coefficient in inductance expansion C0 = Cms (X=0) 0.26399 mm/N constant part in compliance C1 0.0051193 1/N 1st order coefficient in compliance expansion C2 -0.00042359 1/Nmm 2nd order coefficient in compliance expansion C3 -5.3496e-006 1/Nmm^2 3rd order coefficient in compliance expansion C4 4.0003e-007 1/Nmm^3 4th order coefficient in compliance expansion C5 1/Nmm^4 5th order coefficient in compliance expansion C6 1/Nmm^5 6th order coefficient in compliance expansion C7 1/Nmm^6 7th order coefficient in compliance expansion C8 1/Nmm^7 8th order coefficient in compliance expansion K0 = Kms (X=0) N/mm constant part in stiffness K1 -0.081636 N/mm^2 1st order coefficient in stiffness expansion K2 0.0090830 N/mm^3 2nd order coefficient in stiffness expansion K3 -0.00014982 N/mm^4 3rd order coefficient in stiffness expansion K4 -2.9248e-007 N/mm^5 4th order coefficient in stiffness expansion

# Derived Loudspeaker Parameters

For the analysis and synthesis of loudspeaker system it is convenient to use special transducer parameters:

 fs (x) instantaneous resonance frequency of the transducer varying with voice coil displacement QMS(x) mechanical loss factor of the transducer at fs considering driver non-electrical resistances only QES(TV, x) electrical loss factor by considering the electrical resistance RE(TV) only, QT(TV, x) total loss factor at fs and voice coil temperature TV considering mechanical and electrical resistances RMS and RE(TV) only.

In contrast to linear modelling most of these parameters are not constant but depend on the instantaneous state of the transducer (displacement x, the voice coil temperature TV).

# Parameters at the Rest Position

The value of the nonlinear parameters at the rest position (x=0) may be used as input for the traditional linear modelling and may be referred as “linear parameters”. Please note that these parameters depend on the instantaneous state of the driver (voice coil temperature, peak value of displacement) and are presented for three different modes of operation:

## Properties

LARGE+WARM

the transducer is operated in the large signal domain,

the peak value of the displacement is high (|x| < xmax),

the variation of the parameters is not negligible,

the voice coil temperature is increased (D TV > 0) due to heating.

LARGE+COLD

the transducer is operated in the large signal domain,

the peak value of the displacement is high (|x| < xmax),

the variation of the parameters is not negligible,

the effect of heating is compensated while considering the cold voice coil resistance Re(D TV =0).

SMALL SIGNAL

the transducer is operated in the small signal domain,

the amplitude of the excitation signal is sufficiently small,

the displacement is small in comparison to the allowed maximal displacement (|x| << xmax ),

the variations of the nonlinear parameters are negligible,

the increase of voice coil temperature is negligible (D TV » 0),

the effects of the nonlinear, thermal and time-varying mechanisms are negligible,

the transducer behaves almost linear.

 Symbol Large + Warm Large + Cold Small Signal Unit Comment Note: for accurate small signal parameters, use LPM module Delta Tv = Tv-Ta 140 0 0 K increase of voice coil temperature during the measurement Xprot 22.2 22.2 3.1 mm maximal voice coil excursion (limited by protection system) Re (Tv) 9.34 6.10 6.10 Ohm (imported) voice coil resistance considering increase of voice coil temperature Tv Le (X=0) 0.44 0.44 0.31 mH voice coil inductance at the rest position of the voice coil L2 (X=0) 2.93 2.93 1.26 mH para-inductance at the rest position due to the effect of eddy current R2 (X=0) 0.94 0.94 0.94 Ohm resistance at the rest position due to eddy currents Cmes (X=0) 1738 1738 1147 µF electrical capacitance representing moving mass Lces (X=0) 46.37 46.37 33.71 mH electrical inductance at the rest position representing driver compliance Res (X=0) 35.88 35.88 50.27 Ohm resistance at the rest position due to mechanical losses Qms (X=0, Tv) 6.95 6.95 9.27 mechanical Q-factor considering Rms only Qes (Tv) 1.04 0.68 0.98 electrical Q-factor considering Re (Tv) only Qts (X=0, Tv) 0.90 0.62 0.89 total Q-factor considering Re (Tv) and Rms only fs 17.7 17.7 25.6 Hz driver resonance frequency Mms 107.290 107.290 107.290 g (imported) mechanical mass of driver diaphragm assembly including voice-coil and air load Rms (X=0) 1.720 1.720 1.860 kg/s mechanical resistance of total-driver losses Cms (X=0) 0.75 0.75 0.36 mm/N mechanical compliance of driver suspension at the rest position Bl (X=0) 10.37 10.37 10.37 N/A (imported) force factor at the rest position (Bl product) Vas 57.0122 57.0122 27.3643 l equivalent air volume of suspension N0 0.029 0.045 0.045 % reference efficiency (2Pi-sr radiation using Re) Lm 76.8 78.7 78.7 dB characteristic sound pressure level Sd 232.00 232.00 232.00 cm² diaphragm area

# Transducer State

The state information describes the progress of system identification and important transducer variables in the last update interval of the measurement.

 Symbol Value Unit Comment Date 2007-07-21 Time 14:38:46 Serial number 165 Mode Therm: Magnet 6g(7) Record 1963/1963 Laser signal reliable t 02:28:40 h:min:s measurement time Time remaining 00:00:00 h:min:s recalculated at thermal mode(a) Ei (t) 27.6 % error current measurement Ex (t) 10.4 % error laser measurement Eu (t) 100.0 % error amplifier check Delta Tv (Delta Tlim) 139.9 (180.0) K increase of voice coil temperature (limit) Blmin (Bllim) 39.2 (35.0) % minimal force factor ratio (limit) Cmin (Clim) 25.0 (25.0) % minimal compliance ratio (limit) P (Plim) 45.7965 (125.00) W real electrical input power (limit) Lmin 99.9 % minimal inductance ratio Pn 90.971040 W nominal electrical input power P Re 34.146384 W Power heating voice coil Irms 1.912 A rms value of the electrical input current Urms 26.977 V rms value of the electrical voltage at the transducer terminals Ipeak 5.542 A peak value of the electrical input current Upeak 79.121 V peak value of the electrical voltage at the transducer terminals PC 3.70 dB thermal power compression factor Glarge (Gmax) 16.5 (26.0) dB gain of the excitation amplitude increased in the large signal domain (maximum) Mech. system abs. import used to identify mechanical system in absolute quantities Xdc -0.0 mm dc component of voice coil excursion measured in the last update intervall Xpeak 0.7 mm positive peak value of voice coil excursion measured in the last update intervall Xbottom -0.7 mm negative peak value (bottom) of voice coil excursion measured in the last update intervall Xp+ 16.3 mm upper limit of displacement range (99% probability) Xp- -16.5 mm lower limit of displacement range (99% probability) Xprot 22.2 mm maximal voice coil excursion allowed by protection system v rms 0.011 m/s voice coil velocity Db 0.1 % distortion factors representing contribution of nonlinear force factor Dl 0.5 % distortion factor representing contribution of nonlinear inductance Dc 0.0 % distortion factor representing contribution of nonlinear compliance R th total 4.10 K/W Delta Tv / P Re

Voice Coil Temperature D TV(t) and Power P(t)

The increase of the voice coil temperature  D TV(t) in comparison to the electric input power P(t) versus measurement time t shows the thermal characteristic of the transducer.

The different modes of operation can easily be identified in these time plots.

In the Amplifier Mode 1(7) the loudspeaker is disconnected and the gain, polarity and distortion of the power amplifier is checked. Here the amplitude of the current is zero.

In the Resistance Mode 2(7) the dc-resistance of the voice coil is measured by a pulsed noise signal.

In the Linear Mode 3(7) the loudspeaker is connected and noise at low amplitude is used as stimulus. The transducer is operated in the small-signal domain. The temperature of the voice coil at the end of this phase is used as reference temperature TA which equals the ambient temperature.

In the Fast Mode 4(7) the amplitude of the stimulus is increased and the limits of the allowed working range are detected automatically.  The voice coil temperature TV increases with the input power. Both state signals are used as protection variables and are compared with the limit values Plim and Tlim defined by the user.

In the Nonlinear Mode 5(7) the learning speed is reduced and the nonlinear curves are measured at highest precision.

The Thermal Mode 6(7) consist of special heating and cooling phases to identify the thermal parameters

## Displacement x(t)

The displacement signal versus measurement time is represented as

 xpeak(t) positive peak of the voice coil displacement in the update interval, xdc(t) averaged dc-value in voice coil excursion, xbottom(t) negative peak value (bottom value) of the voice coil displacement in the updated interval, xdcmax(t) maximal dc-value in voice coil excursion xdcmax(t)=(xpeak(t)+xbottom(t))/2

Asymmetrical nonlinearities produce not only second- and higher-order distortions but also a dc-part in the displacement by rectifying low frequency components.

For an asymmetric stiffness characteristic the dc-components moves the voice coil for any excitation signal in the direction of the stiffness minimum.

For an asymmetric force factor characteristic the dc-component depends on the frequency of the excitation signal. A sinusoidal tone below resonance (f<fS) would generate or force moving the voice coil always in the force factor maximum. This effect is most welcome for stabilizing voice coil position. However, above the resonance frequency (f>fS) would generate a dc-component moving the voice coil in the force factor minimum and may cause severe stability problems.

For an asymmetric inductance characteristic the dc-component moves the voice coil for any excitation signal in the direction of the inductance maximum.

Please note that the dynamically generated DC-components cause interactions between the driver nonlinearities. A optimal rest position of the coil in the gap may be destroyed by an asymmetric compliance or inductance characteristic at higher amplitudes. The module "Large Signal Simulation (SIM)" allows systematic investigation of the complicated behavior.

# Distortion Analysis

The Distortion Analysis shows the contribution of each nonlinearity to the total distortion in the reproduced output signal for the audio-like excitation signal used during parameter measurement. The identified digital model of the transducer makes it possible to measure the peak value of  the distortion components generated by force factor, compliance and inductance and to relate each value to the peak value of the total output signal (sound pressure):

 db relative degree of distortion generated by nonlinear force factor Bl(x) dL relative degree of distortion generated by nonlinear inductance Le(x) dC relative degree of distortion generated by nonlinear compliance Cms(x) dl(i) relative degree of distortion generated by nonlinear inductance Le(i) – permeability nonlinearity (flux modulation)

The distortion analysis is performed simultaneously with the parameter identification. The relative degrees of distortion are expressed in percent and  presented versus measurement time (in seconds) for the transducer under test:

Each degree is a one-number representation of the distortion summarizing all of the harmonic and intermodulation components. Please note that the amount of distortion depend on the spectral properties of the excitation signal.

# Remedies for Transducer Nonlinearities

## Bl Symmetry xb(x)

This curve shows the symmetry point in the nonlinear Bl-curve where a negative and positive displacement x=xpeak will produce the same force factor

Bl(xb(x) + x) = Bl(xb(x) – x).

If the shift xb(x) is independent on the displacement amplitude x then the force factor asymmetry is caused by an offset of the voice coil position and can be simply compensated.

If the optimal shift xb(x) varies with the displacement amplitude x then the force factor asymmetry is caused by an asymmetrical geometry of the magnetic field and can not completely be compensated by coil shifting.

## Kms Symmetry xc(x)

This curve shows the symmetry point in the nonlinear compliance curve where a negative and positive displacement x=xpeak will produce the same compliance value

kms(xc(x) + x) = kms(xc(x) – x).

A high value of the symmetry point xc(x) at small displacement amplitudes x » 0 indicates that the rest position does not agree with the minimum of the stiffness characteristic. This may be caused by an asymmetry in the geometry of the spider (cup form) or surround (half wave).

A high value of the symmetry point xc(x) at maximal displacement x» xmax may be caused by asymmetric limiting of the surround.