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

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Comments

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

 More information about optimising these parameters can be found in the article Displacement limits

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 K_MS(x) describes the mechanical properties of the suspension. It's inverse is the compliance C_MS(x)

More information regarding Kms(x) and its optimization can be found in the article Adjusting Mechanical Suspension

The inductance components L_e (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 X_BL, X_C, X_L and X_d describe the limiting effect for the force factor Bl(x), compliance C_ms(x), inductance L_e(x) and Doppler effect, respectively, according to the threshold values Bl_min, C_min, Z_max and d₂ used. The thresholds Bl_min= 82 %, C_min=75 %_, Z_max=10 % and d₂=10% generate for a two-tone-signal (f₁=f_s, f₂=8.5f_s) 10 % total harmonic distortion and 10 % intermodulation distortion. The thresholds Bl_min= 70 %, C_min=50 %_, Z_max=17 % create 20 % total harmonic distortion which is becoming the standard for acceptable subwoofer distortion thresholds.

These parameters are defined in more detail in the papers: AN04 Measurement of Peak Displacement Xmax, AN05 - Displacement Limits due to Driver Nonlinearities. , AN17 - Credibility of Nonlinear Parameters, Prediction of Speaker Performance at High Amplitudes, Assessment of Voice Coil Peak Displacement Xmax, and Assessing Large Signal Performance of Loudspeakers

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 T_V).

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:

Mode	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 T_V(t) and Power P(t)

The increase of the voice coil temperature D T_V(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 T_A 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 T_V increases with the input power. Both state signals are used as protection variables and are compared with the limit values P_lim and T_lim 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<f_S) 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>f_S) 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

You can find a detailed description of these non-linearities and their remedies in the papers Loudspeaker Nonlinearities - Causes and Symptoms, Assessing Large Signal Performance of Loudspeakers, and Diagnosis and Remedy of Nonlinearities

Bl Symmetry x_b(x)

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

Bl(x_b(x) + x) = Bl(x_b(x) – x).

If the shift x_b(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 x_b(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.

K_ms Symmetry x_c(x)

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

_kms(x_c(x) + x) = k_ms(x_c(x) – x).

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