HTML Report - Klippel GmbH

Transducer Measurement	KLIPPEL
Distortion Analyzer 1	acoustics and signal processing

Detailed Report:
	Name of Transducer	P 205
	Driver Comment:
	Measurement:	Large Signal Parameter
	Measurement Comment:
	Date:	02/29/00
	Time:	13:21:51
	Username:	wolfgang

Nonlinear Parameters

The dominant nonlinearities are modeled by variable parameters such as

b(x) instantaneous electrodynamic coupling factor (force factor of the motor) defined by the integral of the magnetic flux density B over voice coil length l,
K_MS(x) mechanical stiffness of driver suspension
C_MS(x,t) mechanical compliance of driver suspension (the inverse of stiffness K_MS(x,t)),
L_E(x) part of voice coil inductance which is independent on frequency,
C_MES(x) electrical capacity representing moving mass,
R_ES(x) electrical resistance due to driver suspension losses
L_CES(x) electrical inductance representing driver compliance

depending on the instantaneous voice coil displacement x.

A solid line represents the used working range x_-peak < x < x_+peak between the minimal and maximal peak displacement occurred in last update interval of the measurement. The dotted line shows the allowed working range x_max < x < x_max identified by the automatic gain adjustment by using predefined limit values.

Power Series Expansion

The nonlinear parameters force factor, compliance, stiffness and inductance are expanded in a power series expansion:

Nonlinear Coefficients
Symbol	Number	Unit	Comment
b_0 = b(x=0)	5.376477	N/A	constant part in force factor
b_1	0.463577	N/Amm	1st order coefficient in force factor expansion
b_2	-0.164177	N/Amm^2	2nd order coefficient in force factor expansion
b_3	-0.023253	N/Amm^3	3rd order coefficient in force factor expansion
b_4	0.002862	N/Amm^4	4th order coefficient in force factor expansion
b_5	0.000629	N/Amm^5	5th order coefficient in force factor expansion
b_6	-0.000021	N/Amm^6	6th order coefficient in force factor expansion
b_7	-0.000008	N/Amm^7	7th order coefficient in force factor expansion
b_8	-0.000000	N/Amm^8	8th order coefficient in force factor expansion

l_0 = L_E(x=0)	0.979066	mH	constant part in inductance
l_1	-0.046646	mH/mm	1st order coefficient in inductance expansion
l_2	-0.005001	mH/mm^2	2nd order coefficient in inductance expansion
l_3	0.000184	mH/mm^3	3rd order coefficient in inductance expansion
l_4	0.000082	mH/mm^4	4th order coefficient in inductance expansion
l_5	-0.000022	mH/mm^5	5th order coefficient in inductance expansion
l_6	-0.000005	mH/mm^6	6th order coefficient in inductance expansiont
l_7	0.000001	mH/mm^7	7th order coefficient in inductance expansion
l_8	0.000000	mH/mm^8	8th order coefficient in inductance expansion

c_0 = C_MS(x=0)	2.015241	mm/N	constant part in compliance
c_1	0.126997	1/N	1st order coefficient in compliance expansion
c_2	-0.038797	1/Nmm	2nd order coefficient in compliance expansion
c_3	0.007982	1/Nmm^2	3rd order coefficient in compliance expansion
c_4	-0.000719	1/Nmm^3	4th order coefficient in compliance expansion
c_5	-0.000760	1/Nmm^4	5th order coefficient in compliance expansion
c_6	0.000018	1/Nmm^5	6th order coefficient in compliance expansion
c_7	0.000015	1/Nmm^6	7th order coefficient in compliance expansion
c_8	0.000000	1/Nmm^7	8th order coefficient in compliance expansion

x_pse	4.8	mm	-x_pse < x < x_pse, range where power series is fitted

This representation uses a minimal set of parameters and simplifies the export of the nonlinear parameters to numerical simulations.

Derived Loudspeaker Parameters

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

f_sresonance frequency of the transducer
Q_MS(x) mechanical loss factor of the transducer at f_s considering driver non-electrical resistances only,
Q_ES(T_V, x) electrical loss factor by considering the electrical resistance R_E(T_V) only,
Q_T(T_V, x) total loss factor at f_s and voice coil temperature T_V considering mechanical and electrical resistances R_MS and R_E(T_V) only.

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

Linear Parameters

the value of the displacement varying parameters at the rest position (x=0) is used as input parameters for traditional linear modeling.

The LINEAR PARAMETERS are presented for three different modes of operation:

LARGE+WARM

the transducer is operated in the large signal domain,
the peak value of the displacement is high (|x| < x_max),
the variation of the parameters is not negligible,
the voice coil temperature is increased (D T_V > 0) due to heating.

LARGE+COLD

the transducer is operated in the large signal domain,
the peak value of the displacement is high (|x| < x_max),
the variation of the parameters is not negligible,
the effect of heating is compensated while considering the cold voice coil resistance R_E(D T_V =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| << x_max ),
the variations of the nonlinear parameters are negligible,
the increase of voice coil temperature is negligible (D T_V » 0),
the effects of the nonlinear, thermal and time-varying mechanisms are negligible,
the transducer behaves almost linear.

Linear Parameters
Symbol	Large + Warm	Large + Cold	Small Signal	Unit	Comment
dT_V = T_V-T_A	9	0	0	K	increase of voice coil temperature during the measurement
x_max	4.8	4.8	1.5	mm	maximal value of voice coil excursion (limit)

R_E(T_V)	5.31	5.11	5.25	Ohm	voice coil resistance considering increase of voice coil temperature T_V
L_E(x=0)	0.98	0.98	0.91	mH	voice coil inductance at the rest position of the voice coil
L_2(x=0)	1.29	1.29	1.08	mH	para-inductance at the rest position due to the effect of eddy current
R_2(x=0)	2.42	2.42	2.04	Ohm	resistance at the rest position due to eddy currents
C_MES(x=0)	196	196	199	µF	electrical capacitance representing moving mass
L_CES(x=0)	58.25	58.25	37.76	mH	electric inductance at the rest position representing driver compliance
R_ES(x=0)	44.92	44.92	49.53	Ohm	resistance at the rest position due to mechanical losses

M_MS	5.68	5.68	5.52	g	mechanical mass of driver diaphragm assembly including voice-coil and air load
R_MS(x=0)	0.64	0.64	0.56	kg/s	mechanical resistance of total-driver losses
C_MS(x=0)	2.02	2.02	1.36	mm/N	mechanical compliance of driver suspension at the rest position
b(x=0)	5.38	5.38	5.27	N/A	force factor at the rest position(Bl product)

Q_EPS(x=0, T_V)	0.35	0.34	0.43		electrical Q-factor considering Z_L(f_s, T_V) only
Q_TP(x=0, T_V)	0.31	0.30	0.38		total Q-factor considering all losses
Q_MS(x=0, T_V)	2.61	2.61	3.59		mechanical Q-factor considering R_MS only
Q_ES(T_V)	0.31	0.31	0.38		electrical Q-factor considering R_E(T_V) only
Q_T(x=0, T_V)	0.28		0.34		total Q-factor considering R_E(T_V) and R_MS only
f_s	47.0	47.0	58.1	Hz	driver resonance frequency
V_AS				l	equivalent air volume of suspension
n_0				%	reference efficiency of electroacoustical conversion (2Pi-sr radiation load)
L_m				dB	characteristic sound pressure level

R_TV				K/W	thermal resistance of path from coil to magnet structure
R_TV + R_TM	4.238	4.238	4.238	K/W	total thermal coil resistance
R_TM				K/W	thermal resistance of magnet structure to ambient air
C_TV				J/K	thermal capacitance of voice coil and nearby surroundings

Temporal Variations of the Stiffness K_MS(t, x=0)

The instantaneous the state variables and parameters estimated by the system identification are permanently sampled and stored in the database. The properties of the mechanical suspension change with time. There are reversible and non-reversible processes due to creep, aging and relaxation.

stiffness K_MS(t) and resonance frequency f_s(t) at rest position x=0

Temporal Variations of the Voice Coil Resistance R_E(t)

The voice coil resistance R_E(t) varies during the measurement due to heating of the voice coil. This variations affect

electrical loss factor Q_ES(t, x=0)
efficiency h ₀(t)
power compression factor PC is expressed in dB and describes the loss of efficiency in the pass-band of the transducer where the electric resistance R_E dominates the electric input impedance.
reference sound pressure level SPL(t)

electric resistance R_E(t) and electric loss factor Q_ES(t)

sound pressure level SPL(t), efficiency n(t) and thermal power compression PC(t)

Transducer State

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

State
Symbol	Number	Unit	Comment
Step	4 (5)		Thermal Mode 4(5)
t	00:10:34	h:min:s	measurement time

E_i(t)	6.3	%	error current measurement
E_x(t)	10.5	%	error laser measurement
E_u(t)	7.1	%	error amplifier check

dT_V (dT_lim)	9.0 ()	K	increase of voice coil temperature (limit)
b_min (b_lim)	38.9 ()	%	minimal force factor ratio (limit)
C_min (C_lim)	30.3 ()	%	minimal compliance ratio (limit)
P (P_lim)	1.95 ()	W	electric input power (limit)
G_large		dB	gain of the excitation amplitude increased in the large signal domain
V_L	67.7	%	minimal inductance ratio

x_DC	0.0	mm	dc component of voice coil excursion measured in the last update intervall
x_peak	4.8	mm	positive peak value of voice coil excursion measured in the last update intervall
x_bottom	-4.0	mm	negative peak value (bottom) of voice coil excursion measured in the last update intervall
x_max	4.8	mm	maximal value (limit of the allowed working range) of voice coil excursion
i_peak	1.904	A	peak value of the electric input current
u_peak	13.697	V	peak value of the electric voltage at the transducer terminals
PC	-0.29	dB	thermal power compression factor

d_b	32.4	%	distortion factors representing contribution of nonlinear force factor
d_L	7.2	%	distortion factor representing contribution of nonlinear inductance
d_C	22.2	%	distortion factor representing contribution of nonlinear compliance

Voltage Probability Density Function pdf(u)

The probability density function of the voltage pdf(u) reflects the properties of the excitation signal (noise) and of the power amplifier used. If the power amplifier is not limiting and does not generate a DC-component in the output signal the pdf(u) is almost perfectly symmetrical. The positive and negative peak values, the rms-value and the crest-factor of the signal can be derived from the properties of the pdf(u).

Voltage u_peak(t) and Current i_peak(t)

The electric signals at the transducer terminals are represented by

i_peakpeak value of the electric input current,
u_peak peak value of the electric voltage at the transducer terminals.

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

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

increase of voice coil temperature dT_V(t) and electric input power P(t)

The different modes of operation can easily be identified in the time plot.

The low values of both states at the beginning of the measurement correspond with Linear Mode 2(5) where 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.
The increase of the input power indicates the Fast Mode 3(5) where the allowed range of safe operation is identified. Usually 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 Thermal Mode 4(5) the excitation signal is attenuated for 120 s to measure the thermal parameters of the voice coil from the temperature response.
In the Final Mode 5(5) the input power is almost constant but the voice coil temperature T_V may increase to higher values due to the rising temperature of the magnet structure rising with a longer time constant.

Displacement x(t)

The displacement signal versus measurement time is represented as

x_peak(t) positive peak of the voice coil displacement in the update interval,
x_dc(t) averaged dc-value in voice coil excursion,
x_bottom(t) negative peak value (bottom value) of the voice coil displacement in the updated interval,
x_dcmax(t) maximal dc-value in voice coil excursion x_dcmax(t)=(x_peak(t)+x_bottom(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 characterteristic 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 above the resonance frequency (f>f_S) would generate a dc-component moving the voice coil in the force factor minimum but a tone below with (f<f_S) would move the voice coil in the force factor maximum.

Displacement Probability Density Function pdf(x)

The probability density function of the displacement signal pdf(x) depends on the properties of the excitation signal (noise) and on the behavior of the transducer as well:

An asymmetrical pdf(x) indicates a dynamic generation of a dc-component in the displacment caused by asymmetric transducer nonlinearities.
The pdf(x) plays an important role as a weighting function in the nonlinear system identification and shows in which region of the displacement the nonlinear parameters are measured with highest precision.

Distortion Analysis

The Distortion Analysis shows the contribution of each nonlinearity to the total distortion while reproducing a audio-like signal in the maximal range of operation:

d_b relative degree of distortion representing contribution of nonlinear force factor,
d_L relative degree of distortion representing contribution of nonlinear inductance,
d_C relative degree of distortion representing contribution of nonlinear compliance.

Remedies for Transducer Nonlinearities

Optimal Voice Coil Shift x_b(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 the and the asymmetry can only partly compensated.

Optimal Suspension Shift x_c(x)

The optimal suspension shift 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 using a suspension design having an asymmetric geometry either of the spider (pot form) or of the surround (half wave).
The optimal suspension shift x_c(x) at high displacement amplitudes x» x_max indicates an asymmetric limiting of the suspension mostly caused by an offset between the surround and spider position.

(c)11/1999 Klippel GmbH Germany - http://www.klippel.de/