FLSI – Fast Large Signal Identification

FLSI – Fast Large Signal Identification #

Introduction #

Welcome #

Congratulations on your purchase of the Fast Large Signal Identification R&D software module.

Do not hesitate to contact us if you need any assistance on using this Klippel R&D module, if you have suggestions for improvements, feature requests or if you found a problem.

At higher amplitudes, loudspeakers generate substantial distortion in the output signal due to nonlinearities inherent in the transducer. The FLSI identifies these nonlinearities and links them to specific physical mechanisms, design decisions, material properties, and assembly techniques of the transducer. It provides valuable diagnostic insight into the speaker’s physical behavior — crucial for design verification, defect detection, and optimization in terms of sound quality, weight, size, and cost.

The FLSI is the successor to the R&D modules LPM and LSI3.

The primary nonlinearities arise from variations in force factor, mechanical stiffness, inductance, and mechanical resistance, depending on voice coil displacement and velocity. These effects are modeled using a nonlinear lumped parameter model, as shown below:

See the section Lumped parameter modeling for more detailed information about speaker modeling.

Features in short #

Measures small and large signal lumped speaker parameters
Applicable to all types of electrodynamic transducers
Supports drivers in free air, closed, vented, and passive radiator systems
Minimal measurement time: from 20 seconds to 3 minutes
Fully automatic measurement process
Identifies the root causes of distortion
Generates accurate linear and nonlinear parameters for simulations
Enables fast thermal characterization
Creates initial parameters for Klippel Controlled Sound (KCS)

Novel features of the FLSI #

Fast measurement and comparison of multiple speaker samples
- Measurement time as low as 20 seconds for small speakers
Automatic measurement of accurate linear and nonlinear parameters
- Automatic stimulus configuration for both linear and nonlinear measurements
- Full lumped parameter set identified in a single measurement
- No need to import Bl(x=0) from the LPM into the large signal identification
Easier identification of maximum displacement limits using new protection metrics
- Maximum displacement (X target) measured with a laser
- Rub&Buzz protection limit determined using a microphone
Comprehensive distortion analysis
- Instantly view both dominant and negligible sources of nonlinear distortion
- Calculation and display of the spectral distribution of individual distortion components
Improved loudspeaker model
- Enhanced accuracy of all linear and nonlinear parameters
- Frequency-dependent inductance nonlinearity curves
- Mechanical creep identified from electrical data — no laser is required
- Mechanical hysteresis included in the model and shown as a distortion component
- Identification of port nonlinearity distortion
- Identification of effective surface area nonlinearity S_d(x) distortion
Fast thermal parameter identification
- Enables calculation of short-term power handling
- Supports comparison of different motor and coil geometries with respect to power handling and cooling performance

System Requirements #

The following hardware is required for running the FLSI:

KLIPPEL Analyzer 3 (KA3)
Power amplifier or KLIPPEL AMP Card

It is highly recommended to use

Microphone
Triangulation laser

The following software is required for running the FLSI:

An installation of KLIPPEL dB-Lab
FLSI license

Internet access is required for parameter identification on the FLSI Server. See our Privacy Policy below.

In addition, if any software restriction policy is in place on your PC, please make sure that you are permitted to run the software cUrl. This program is used to send the measurement signals to a Klippel server which identifies the speaker model.

dB-Lab installs cUrl under the path

%ProgramData%\Klippel\<DB-LAB NAME>\Scripts5\Klippel\Library\curl\CURL.EXE.

%ProgramData% usually links to C:\ProgramData and <DB-LAB NAME> is usually RnD, if only a single dB-Lab is installed on your computer.

Tutorial: Performing a measurement #

Hardware Configuration #

The hardware setups on the KLIPPEL KA3 comprising one Speaker Card, one Laser Card, one XLR Card, and an optional AMP Card is illustrated in the figures below. For more information about the hardware, please see the hardware manual.

Connect the DUT (device under test) to the SP 1 or SP 2 Speakon connectors of the KA3’s Speaker Card.

Note

Make sure that the the DUT is connected tightly. Alligator clips can corrode over time, especially in humid areas. Also their springs can wear out, which leads to loose connections. The FLSI notices loose connections by analyzing the voltage and current signals and will display an error message. In this case, it might be necessary to replace them.

Laser and microphone sensors are not necessarily required for operating the FLSI. However, these sensors are highly recommended to get a comprehensive set of diagnostics information. The laser measures the voice coil’s peak displacement, which is then used to automatically determine the force factor Bl(x=0) (see Mechanical and Acoustical Calibration). The microphone measures impulsive distortion, which can be used as a protection limit, and nonlinear distortion, which provides information about the harmonic and intermodulation distortion created by the deformation of the surround.

The microphone should be placed in the speaker’s near-field (see Measuring Sd(x)). Make sure that IEPE/phantom power is switched on by pressing the buttons next to the microphone connectors on the KA3’s front panel.

Hardware connection on KA3 with AMP Card

Hardware connection on KA3 without AMP Card

Routing Configuration #

In the dB-Lab software’s tool bar, open the KA3 Signal Configuration to setup the routing. Set the correct output channel (AMP card or XLR Card if an external amplifier is used) and input channels for the optional external sensors.

If not done already, calibrate your laser.

Note

The current sense (Low/High Sensitivity) is set automatically. The setting specified in the Signal Configuration is disregarded.

FLSI Software Configuration #

Open a database by pressing Select Database in the dB-Lab software. Create a new FLSI Fast Large Signal Identification operation, open its Properties by right-clicking on the operation and set up the measurement as follows. It is assumed that the hardware is set up as described in Hardware Configuration and the laser and the microphone are connected to the KA3.

1. Specify the Hardware Configuration#

Select which sensors are connected in the Peripherals category. This is required for the measurement module to decide which protection parameters are available.

In the KA3 Routing category, select which in- and output channels of the hardware device shall be used.

2. Select the Speaker System Type#

See Speaker System Modeling for detailed information.

3. Select the Protection Parameters#

Warning

Wrong settings can damage your driver. Be especially careful with tweeters, micro-speakers and telecommunications drivers. For those, consider decreasing the Start Voltage on the property page, in case the device under test could already be close to its limit with the default value.

Impulsive Distortion

A microphone is required to use this part of the protection system. The impulsive distortion is identified by exploiting out-of-band distortion in the sound pressure signal. The crest factor CID_MT of the out-of-band distortion is calculated. High CID_MT values indicate that the speaker is working above its safe working range.

Note that the impulsive distortion metric CID_MT is calculated similarly to the CID for sinusoidal excitation (see the TRF Transfer Function module). However, the crest factor of the multi-tone is higher than that of the sinusoidal tone. Therefore, multi-tone’s impulsive distortion limit is higher than the sinusoidal excitation’s one.

Note

The Impulsive Distortion limit is very valuable in many cases to determine a safe maximum voice coil displacement. However, if there is background noise, parasitic vibrations e.g. originating from the speaker stand or rattling objects in the room, this protection system will trigger already at low displacements where the speaker might still behave linearly and the nonlinear parameter variations are low. In this case, use other protection parameters such as X target and deactivate the Impulsive Distortion protection.

X target: This parameter specifies the maximum absolute peak voice coil displacement that shall be used in the measurement. The peak displacement value is measured by a laser sensor.

Note

The voice coil’s rest position is measured at the very beginning of the LINEAR MODE. Ensure that the positions of the DUT and the laser remain unchanged throughout the entire measurement.

Relative Bl(x) min: The minimal force factor ratio Bl_min(x) is a useful criterion to detect the maximum displacement x_max of the voice coil. This parameter reflects the ratio of the voice coil height to the thickness of the pole plates in the motor structure. A low minimal force factor ratio, e.g. Bl_min(x) < 50% produces substantial distortion components in the output signal spreading over the whole excited frequency band and indicates the end of the useable working range. The default value might be decreased if the driver is capable and intended to be used in a more nonlinear region. Increase this value if the DUT is fragile or it has a very high voice coil overhung.
Relative C(x) min: The minimal compliance ratio C_min(x) describes the geometrical changes of the suspension. A high decrease of the compliance, e.g. to C_min(x) < 50%, produces substantial harmonic distortion components in the output signal and indicates the end of the useable working range. The default value might be decreased, if the driver is capable and intended to be used in a more nonlinear region.

Note

The working range of tweeters, headphones, micro-speakers and other transducers without spider is usually not limited by a progressive mechanical stiffness. For these speaker types, it is recommended to activate the impulsive distortion limit. In addition, the maximum displacement X_target limit or a conservative manually entered excitation voltage might be used. This ensures that the transducer operates in its permissible working range and avoids damage due to hard limiting.

Coil Temperature: This protection limit defines the maximum allowed relative voice coil temperature increase. If this limit is exceeded, the measurement is immediately stopped to avoid permanent voice coil damage. The measurement of the coil temperature is done by measuring its DC resistance using a low frequency tone f_Re. The reference temperature is measured at the beginning of the LINEAR MODE. Make sure that the voice coil temperature is at approx. room temperature at the beginning of the measurement.

Note

Voice coils do not heat up homogeneously. Usually, the hottest place is at the rims, while it is cooler in the gap. The FLSI measures the average temperature. Hence, allow some margin to the maximum absolute maximum temperature rating of the voice coil.

Starting the Measurement #

Click the Run button.

After the measurement is finished, the measurement results are available.

Accelerating the Measurement #

When performing the first measurement of a speaker model, it is recommended to firstly determine the peak displacement and optimal measurement settings (e.g., excitation voltages) automatically, following the procedure outlined in FLSI Software Configuration. If the speaker is large and can cope with higher powers, the setting Start Voltage (Enlargement Mode) can be increased on the operation’s property page. This will accelerate the ENLARGEMENT MODE, allowing it to start at a higher voltage.

If the thermal parameters of the DUT are known or not relevant, the THERMAL MODE can be deactivated to further reduce measurement time

If the DUT or a device with the same design has been characterized previously, you can switch the LINEAR and NONLINEAR MODE settings from Automatic to Manual. In this case, the automatic level determination in the LINEAR MODE and ENLARGEMENT MODE will be skipped. This is particularly useful when measuring multiple samples of the same speaker model. The protection system, except the thermal protection, is inactive in manual mode. Ensure that the transducer can handle the given voltage.

Note

Tip: If you set the settings to Manual on an already existing operation that holds result data, the property page displays the settings (excitation level, number of averages) of the previous measurement.

Manual Measurement Settings skipping ENLARGEMENT and THERMAL MODE

Theory #

Measurement Technique #

Parameter Identification#

The FLSI identifies both linear and nonlinear parameters, as well as state information of the transducer. In addition to the Thiele-Small parameters that are valid only at small amplitudes, the FLSI identifies parameters that describe the nonlinear behavior over the entire transducer’s working range.

The FLSI measures the free parameters of an extended loudspeaker model using a full dynamic measurement. The loudspeaker is excited with a frequency-shaped multi-tone stimulus. The stimulus bandwidth is determined automatically by measuring the DUT’s resonance frequency, approximately covering the bandwidth relevant to the target application. Additionally, a low-frequency tone is applied to measure the DC resistance, which is necessary for tracking the voice coil temperature required for thermal protection and the identification of the thermal model. Optimal parameters are estimated by a nonlinear system identification algorithm, primarily based on the voltage and current signals available at the loudspeaker terminals.

In addition to measuring the electrical signals, an optional laser sensor should be used to measure displacement and identify the force factor Bl(x=0) (see Mechanical and Acoustical Calibration). A microphone can be used to approximate the nonlinear distortion contribution due to the modulation of the transducer’s effective surface area S_d(x) (see Distortion Components).

The total harmonic and intermodulation distortion of each modeled nonlinear mechanism in the loudspeaker is calculated for the used multi-tone excitation signal. This enables the detection of the physical cause of the dominant distortion.

This information is crucial for identifying the weakest points in the loudspeaker design and providing insights for potential improvements. Some nonlinear distortion can be reduced without increasing the speaker’s cost. For example, distortion caused by asymmetries in the parameter characteristics can be minimized by optimally centering the voice coil and mechanical suspension.

The full nonlinear transducer model serves as the basis for numerical simulations to predict the transducer’s nonlinear behavior in various applications (SIM, SIM‑AUR modules).

Measurement Conditions#

Mounting#

To measure the linear and nonlinear parameters of a loudspeaker driver, it is recommended to first measure it in free air with the loudspeaker axis pointing horizontally to prevent gravity from shifting the voice coil’s rest position. A loudspeaker stand, such as the Klippel Pro Driver Stand or Klippel Micro-Speaker Clamping, simplifies mounting and provides easy access for the triangulation laser.

If a large driver mounted in an enclosure needs to be measured, a stand may be impractical due to its limited size and the potential for rattling, as significant forces are involved. In this case, the loudspeaker can be fixed to the floor and decoupled from the laser sensor. Ensure that the speaker remains stationary during the measurement.

Loudspeaker systems#

Transducers must be measured under conditions that ensure the mechanical system, including any acoustical load, can be represented as a second- or fourth-order system (see Speaker System Modeling). Higher order systems are currently not supported.

It is recommended to measure the DUT in free air, as this requires less electrical input power and produces a lower sound pressure level compared to measuring the same driver in a closed system.

However, the FLSI allows for the measurement of parameters in closed and fourth-order speaker systems, such as vented, bandpass, or passive radiator systems. This is especially useful for determining the dominant nonlinear distortion components in a loudspeaker system with an enclosure similar to that used in the final product. Additionally, the FLSI can identify enclosure parameters, such as the linear parameters of the full fourth-order system (e.g., port resonance frequency and Q-factor), or the nonlinear distortion caused by the port.

Measuring Sd(x)#

The effective radiation area S_d(x=0) can vary over the voice coil displacement x due to the deformation of the surround (see Mechano-acoustical transduction). The nonlinear distortion caused by the variation in S_d(x) is approximated by exploiting information found in the sound pressure signal, which is measured by a microphone. For the highest precision, the distance from the microphone to the transducer’s cone should be approximately 1 to 2 times the transducer’s diameter.

It is possible to calculate the S_d(x) distortion in a free-air measurement. However, the sound pressure signal is linearly distorted by the acoustical shortcut between the front and the rear of the transducer’s cone and surround. To obtain the S_d(x) distortion spectrum that corresponds as closely as possible to the final application, we recommend performing the measurement with the transducer mounted in an enclosure.

Measurement Procedure#

The measurement procedure is organized as follows.

Measurement Procedure#
Step	Operation Mode	Comment
1	LINEAR MODE	Measurement of the linear parameters in the small signal domain
2	ENLARGEMENT MODE	Identification of the allowed working range
3	NONLINEAR MODE	Identification of the nonlinear parameters
4	THERMAL MODE	Identification of thermal parameters

LINEAR MODE:

Before driving the loudspeaker with an excitation signal, all equipment is checked for potential issues, such as

correctly connected device under test
amplifier gain
inner amplifier resistance

If any issues are detected, an error message is displayed, and the measurement is canceled.

Next, multiple measurements are performed to determine the optimal small-signal stimulus settings (excitation voltage, number of averages, signal bandwidth). The small-signal parameters are identified based on the measured voltage, current, and displacement (optional) signals.

ENLARGEMENT MODE (optional):

After a successful linear parameter estimation, the nonlinear parameters are estimated in the large signal domain by gradually increasing the amplitude of the excitation signal until one of the protection criteria reaches its predefined limit.

The ENLARGEMENT MODE is divided into two parts: The first part uses very short stimuli to roughly determine the transducer’s limit. Once a protection limit is nearly reached or exceeded, the second part of the ENLARGEMENT MODE begins. A longer stimulus, more similar to the NONLINEAR MODE’s stimulus, is applied to precisely determine the transducer’s physical operating limit.

If the protection limits Bl_min or C_min (see 3. Select the Protection Parameters) are active, a fast nonlinear parameter identification is performed after each measurement step. Note that the resulting parameters may not have the highest precision, as this mode prioritizes maximum speed.

NONLINEAR MODE (optional):

After determining the optimal range of operation, the nonlinear parameters are identified using the optimal stimulus settings (excitation voltage, bandwidth) with the highest accuracy.

THERMAL MODE (optional):

This mode identifies the short-term thermal parameters. The voice coil is excited by a multi-tone signal with high power until its temperature is sufficiently high to reliably determine the thermal parameters. If the voice coil was already sufficiently heated during the NONLINEAR MODE, this heating phase might be skipped. Afterward, the voice coil is cooled down. Based on the heating and cooling characteristics, the thermal parameters are identified.

Lumped parameter modeling #

Linear and nonlinear loudspeaker behavior produces substantial linear and nonlinear distortion in the output signal. These effects are closely related to the transducer’s design principles, material properties, and assembly techniques, and can be predicted using nonlinear modeling.

The FLSI R&D module combines linear and nonlinear measurement techniques with nonlinear physical simulation algorithms. It explains the generation of distortion, identifies the underlying physical causes, and provides suggestions for constructional improvements. This information helps to better understand the linear and nonlinear mechanisms acting in the loudspeaker, enabling further optimization of the speaker with respect to sound quality, weight, size, and cost.

At low frequencies, an electro-mechanical equivalent circuit with lumped elements can successfully model electrodynamic transducers. The structure of this model represents the transducer principle and the basic physical mechanisms within the transducer. The free parameters of the model vary depending on:

the transducer type,
unit-to-unit variations, and
from state quantities such as:
- electrical current,
- voice coil displacement,
- voice coil velocity,
- ambient temperature,
- mounting conditions,
- the acoustic sound field and
- the electrical excitation signal.

Small-Signal Modeling #

The linear electrodynamic transducer model provides a simplified description of moving-coil loudspeakers under the assumption of small-signal operation — all parameters are considered constant and independent of displacement, velocity, current, and temperature. The model couples an electrical subsystem to a mechanical subsystem through an ideal electromechanical transformer.

Electrical Subsystem#

The electrical domain is described by the voice-coil resistance and inductance. Applying Kirchhoff’s voltage law yields

\[u(t) = R_\mathrm{e} i(t) + L \frac{\mathrm{d} i(t)}{\mathrm{d} t} + Bl\, v(t),\]

where \(u(t)\) is the applied terminal voltage, \(i(t)\) the voice-coil current, \(R_\mathrm{e}\) the DC resistance, \(L\) the voice-coil inductance, \(Bl\) the force factor (assumed constant in the linear model), and \(Bl\, v(t)\) the back electromotive force (EMF).

Mechanical Subsystem#

The mechanical domain is modeled as a lumped mass–spring–damper system driven by the Laplace (Lorentz) force

\[F(t) = Bl\, i(t).\]

The equation of motion of the diaphragm and voice-coil assembly is

\[M_\mathrm{ms} a(t) + R_\mathrm{ms} v(t) + K_\mathrm{ms} x(t) = Bl\, i(t),\]

where the parameters are the moving mass \(M_\mathrm{ms}\), the mechanical resistance (losses) \(R_\mathrm{ms}\), and the mechanical stiffness \(K_\mathrm{ms}\). The states are the displacement \(x(t)\), the velocity \(v(t) = `\dot{x}(t)\), and the acceleration \(a(t) = \ddot{x}(t)\).

Frequency-Domain Representation#

In the frequency domain (\(s = j\omega\)), the mechanical impedance is

\[Z_{\mathrm{m}}(s) = R_{\mathrm{ms}} + s M_{\mathrm{ms}} + \frac{K_{\mathrm{ms}}}{s}.\]

The electrical input impedance of the transducer becomes

\[Z_{\mathrm{e}}(s) = R_\mathrm{e} + s L + \frac{(Bl)^2}{Z_{\mathrm{m}}(s)}.\]

This expression illustrates the electromechanical coupling: mechanical resonance and losses are reflected into the electrical domain via the squared force factor.

Large-Signal Modeling #

The linear model assumes small excursions, constant force factor, linear suspension stiffness, linear damping, and negligible magnetic saturation and thermal effects. Despite these simplifications, it is very valuable for loudspeaker analysis and linear system design. However, significant deviations from this model at higher signal levels motivate the introduction of nonlinear and time-varying extensions:

Self-Inductance#

For an ideal coil, the inductance can be modeled using a constant inductance parameter \(L\). The electrical impedance of the ideal inductance is

\[\underline{Z}_\mathrm{L,ideal}(f) = 2\pi f j L.\]

However, since the voice coil is surrounded by permeable and conductive materials, its self-inductance cannot be modeled as a simple, ideal inductance. This is due to iron losses arising from eddy currents, the skin effect, and magnetic hysteresis. Additionally, conductive materials in the motor, such as copper caps or shorting rings, reduce the self-inductance depending on frequency. As a result, the linear complex impedance of the lossy inductance, valid in the small signal domain, becomes frequency-dependent and comprises a reactive part \(L(f)\) and a resistive part \(R_\mathrm{L}(f)\) (both are real-valued, frequency-dependent parameters).

\[\underline{Z}_\mathrm{L,linear}(f) = 2\pi f j L(f) + R_\mathrm{L}(f).\]

Section Lossy Inductance explains how the small signal lossy inductance is displayed in FLSI.

In addition, the magnetic flux path changes depending on the voice coil’s position: When the voice coil moves deeper into the gap toward the back plate, it becomes more surrounded by iron, and the magnetic field lines pass predominantly through this high-permeability material. This increases the inductance and iron losses. Conversely, when the coil moves outward (away from the backplate) a larger portion of the magnetic field lines passes through air, which has a relative permeability close to unity. This reduces the inductance and iron losses.

The self-inductance also varies with the electrical current \(i(t)\) due to the nonlinear magnetization behavior of the surrounding ferromagnetic material (iron) in response to the AC magnetic field (see Flux Modulation).

Hence, the impedance of the inductance is frequency-dependent and nonlinear (also see Lossy Inductance (x)):

\[\underline{Z}_\mathrm{L}(f,x,i) = 2\pi f j L(f,x,i) + R_\mathrm{L}(f,x,i).\]

In addition to the high-frequency intermodulation distortion caused by this nonlinear behavior, the non-uniform distribution of the alternating magnetic field along the voice coil’s axis of motion generates a reluctance force \(F_\mathrm{r}\). For an ideal, lossless inductor described by the impedance \(\underline{Z}_\mathrm{L,ideal}(x)\) which depends solely on displacement, this force can be calculated as

\[F_\mathrm{r,ideal}(x, i) = \frac{{i\left(t \right)}^{2}}{2}\frac{\partial L\left(x \right)}{\partial x}.\]

The nonlinear distortion generated by this force is displayed in Distortion Components.

The reluctance force can cause a significant DC displacement, shifting the voice coil towards the highest nonlinear inductance \(L(f, x)\). The DC forces generated by nonlinearities such as the reluctance force are displayed in the table Nonlinear Parameters under Distortion Components.

Nonlinear electro-mechanical transduction#

As explained in Small-Signal Modeling, the electro-mechanical transduction parameter (force factor) converts the current into the Laplace force. \(Bl\) was assumed to be constant in the small-signal modeling. However, in the large-signal domain, it can vary significantly with voice coil displacement \(x\) (see Force Factor Bl(x)). The shape of the nonlinearity depends on the voice coil-gap configuration (i.e., overhung, underhung, equal length).

Hence, the Laplace force becomes

\[F(t) = Bl(x) i(t).\]

Also the back EMF gets distorted by \(Bl(x)\):

\[u_\mathrm{emf} = Bl(x) v(t)\]

Mechanical Resonator#

The Laplace force acts on a resonator that consists of

the moving mass \(M_\mathrm{ms}\),
the now nonlinear, time-variant, mechanical stiffness \(K_\mathrm{ms}(t, f, x)\), and the
nonlinear, time-variant mechanical resistance \(R_\mathrm{ms}(t, f, v, F)\).

While the moving mass is expected to be constant with respect to displacement, velocity, time, frequency, and temperature, the suspension’s mechanical stiffness and resistance can fluctuate significantly:

\(K_\mathrm{ms}\) and \(R_\mathrm{ms}\) are frequency-dependent due to mechanical creep (see Mechanical Stiffness Km(f,x=0)).
\(K_\mathrm{ms}\) depends on the voice coil displacement \(x\) (see Mechanical Stiffness Km(x)) due to the stretching of the spider’s corrugation rolls and the surround.
\(K_\mathrm{ms}\) and \(R_\mathrm{ms}\) are highly time-variant due to the deformation of the suspension parts and mechanical losses. They can vary significantly between the small and large signal domain (see Linear Parameters at x=0).
\(R_\mathrm{ms}\) depends on the voice coil velocity \(v\) due to the air flow resistance** in the motor structure (relevant in smaller transducers). See the result curve Mechanical Resistance Rms(v).
\(R_\mathrm{ms}\) depends on the excitation force and time (\(F\) and \(t\)) due to internal damping caused by nonlinear creep and mechanical hysteresis. The nonlinear distortion created by this effect can be viewed in Distortion Components.

Mechano-acoustical transduction#

The transducer cone with an effective surface area \(S_\mathrm{d}(x)\) converts the velocity into a volume velocity

\[q(t) = S_\mathrm{d}(x)v(t)\]

and generates a sound pressure \(p(t)\). The effective diameter \(S_\mathrm{d}(x)\) can vary with voice coil displacement \(x\), due to the deformation of the surround. This nonlinearity can be dominant when the surround’s surface area is large compared to the cone’s surface area. This is often the case in small transducers (i.e., micro-speakers) and long-throw woofers. The distortion generated by \(S_\mathrm{d}(x)\) can be found under Distortion Components. \(S_\mathrm{d}(x)\) distortion is identified using a microphone; further information about the measurement conditions can be found here.

Acoustic Load#

If the transducer is mounted in an enclosure instead of being operated in free air, a load impedance Z_load must be considered, as this changes the total system response. This load impedance can be nonlinear and generate additional nonlinear distortion. E.g., a port in a vented box will create nonlinear distortion due to the nonlinear air flow resistance. In passive radiators, the nonlinear suspension stiffness will create additional nonlinear distortion at low frequencies, comparable with the nonlinear suspension suspension of electro-dynamic transducers. See Speaker System Modeling for acoustic load modeling.

Thermal model #

In addition to the lumped parameters of the electrical, mechanical, and acoustical domain, FLSI identifies the parameters of the short-term thermal model displayed below.

A parameter identification algorithm determines the thermal parameters of the transducer:

The thermal capacitance \(C_\mathrm{tv}\) is primarily related to the mass of the voice coil, as it reflects the coil’s ability to store thermal energy.
The thermal resistance \(R_\mathrm{tv}\) is mainly associated with the surface area of the voice coil, which determines the rate of heat dissipation through conduction and radiation.
The effective thermal resistance \(R_\mathrm{tc,eff}\), defined as the RMS value of the time-variant and nonlinear resistance \(R_\mathrm{tc} (t,x,v)\) for the given stimulus, accounts for convective cooling effects. It depends on the surface area of the voice coil, the voice-coil position, the air velocity in the gap, and the temporal evolution of these quantities.

This model is designed to provide reliable and repeatable results for any type of electro-dynamic transducer within a short measurement time. This is particularly valuable for:

quickly estimating the voice coil’s power handling capabilities,
experimenting with different voice coil and motor geometries,
experimenting with different voice coil former materials,
evaluating the effect of convection cooling — if \(R_\mathrm{tc,eff} \gg R_\mathrm{tv}\) for the given stimulus, convection cooling is negligible.

Note

The linear parameters of the thermal model, \(C_\mathrm{tv}\) and \(R_\mathrm{tv}\), are independent of the measurement configuration.

However, to ensure accurate parameter identification, it is essential that the voice coil temperature is approximately at room temperature at the beginning of the measurement. This implies that when repeating a measurement (including THERMAL MODE) on the same DUT, a cool-down period may be necessary. For micro-speakers, a pause of about one minute or less is typically sufficient. In contrast, for large transducers such as subwoofers, a longer cool-down time of up to 10 minutes may be required. The temperature state of the voice coil can be verified by running only the LINEAR MODE in FLSI: if the measured small-signal DC resistance \(R_\mathrm{e}\) is within 2% of the initial value (measured on a cold voice coil), the full measurement can proceed.

Unlike \(C_\mathrm{tv}\) and \(R_\mathrm{tv}\), the effective thermal resistance \(R_\mathrm{th,eff}\) depends on the stimulus — it is influenced by state variables such as voice coil velocity and displacement, as these affect convective cooling. Consequently, comparisons of \(R_\mathrm{th,eff}\) between different DUTs are only meaningful if the RMS displacement and velocity are similar across measurements. This can be ensured, for instance, by manually adjusting the stimulus level in the FLSI property page.

The relative temperature

\[\Delta T_\mathrm{v} (t) = T_\mathrm{v} (t) - T_\mathrm{a}\]

is the difference between the absolute mean voice coil temperature \(T_\mathrm{v}(t)\) and the ambient temperature \(T_\mathrm{a}\). It is approximated using the measured change of the voice coil DC resistance \(R_\mathrm e\).

\[\Delta T_\mathrm{v}(t) = \frac{\frac{R_\mathrm{e}(t)}{R_\mathrm{e}(t=0)} - 1}{\alpha_\mathrm{T}}\]

FLSI uses a fixed temperature coefficient of

\[\alpha_\mathrm{T} = 0.0038 \frac{1}{K}.\]

Example:

The plot below shows typical measured (green) and simulated (red) voice coil temperature increase of a small woofer over a time of 1 minute.

Modeled and Measured Voice Coil Temperature with constant power dissipation of 5 W

In this example, the following assumptions are made:

In the beginning at \(t=0\), the voice coil temperature is identical to the ambient temperature \(T_\mathrm{a}\).
The voice coil is excited with a broadband signal.
The power dissipated in the conductive voice coil material is assumed to be constant over time at \(P_\mathrm{Re} = 5 W\).
The RMS velocity, \(v_\mathrm{rms}\), is constant.

It can be observed that the modeled temperature behaves similarly to the measured temperature in the beginning of the measurement, until approx. 3 times the effective total time constant

\[\tau_\mathrm{h,eff} = \left(R_\mathrm{tc,eff} || R_\mathrm{tv}\right) \cdot C_\mathrm{tv}.\]

This information is very useful for estimating a rough worst-case short-term power handling capability of the voice coil (neglecting convection cooling), using the relationship

\[P_\mathrm{max,short} = \frac{\Delta T_\mathrm{v,max}}{R_\mathrm{tv}},\]

where \(\Delta T_\mathrm{v,max}\) is the maximum permissible voice coil temperature increase, determined by the specifications of the voice coil and the voice coil former.

Note

The short-term thermal model is not designed to identify parameters to be used for long-term simulation. Complex models (e.g. Nonlinear Modeling of the Heat Transfer in Loudspeakers) have been developed, that allow to predict the voice coil temperature over a long time with any input signal. However, to identify the free parameters of such complex models, a much longer measurement time is required. In addition, the model might have to be adjusted, depending on the type of speaker, e.g. modeling a subwoofer requires a different model than a smartphone speaker due a different motor layout.

Speaker System Modeling #

The following speaker system type models are currently supported and used. They can be selected on the FLSI’s property page.

Free Air

An electrodynamic transducer without enclosure. The mechanical system is described by a second-order system (one distinct resonance peak in the electrical impedance magnitude).

Closed box

An electrodynamic transducer mounted in a closed enclosure. The mechanical system is described by a second-order system (one distinct resonance peak in the electrical impedance magnitude).

The compliance C_ab(x) represents the compliance of the enclosed air in the rear chamber. The chamber’s volume is modulated depending on the cone’s position. This causes the air compliance to be nonlinear.

Vented box

Also referred to as bass reflex system. An electrodynamic transducer mounted in an enclosure incorporating an opening such as a pipe. Together, they act as a Helmholtz resonator that is excited by the transducer. The mechanical system is described by a fourth-order system (two distinct resonance peaks in the electrical impedance magnitude).

At higher excitation levels, the air particle velocities in and around the port can become very high. This leads to high acoustical losses due to vortex shedding and laminar and turbulent air flow resistance. These effects can create high nonlinear distortion. In addition, an asymmetric nonlinear port resistance (see Nonlinear Parameters) can create a significant undesired voice coil DC displacement.

Passive radiator

An electrodynamic transducer mounted in an enclosure drives a passive membrane (spring-mass system) that is mounted in the same enclosure. The mechanical system is described by a fourth-order system (two distinct resonance peaks in the electrical impedance magnitude).

Similarly to the nonlinear transducer suspension (see Mechanical Resonator), the passive radiator’s suspension stiffness and resistance cannot assumed to be linear and constant. It changes over time and displacement.

Sidefire: A small electrodynamic transducer mounted in an enclosure which is closed on the rear side of the transducer’s membrane. On the front side it has a small sound output opening. Such speakers are e.g. used in mobile phones. The linear model of this system type is identical to a fourth-order bandpass system. Its mechanical system is usually described by a fourth-order system (two resonance peaks in the electrical impedance magnitude). If only a single distinct resonance peak in the electrical impedance is found, the system is described as a second-order system.
Tweeter: An electrodynamic transducer used to reproduce high frequencies. Usually, the voice coil is directly attached to a dome-shaped cone. This assembly acts on a closed rear cavity that might include a small internal port. The mechanical system can be described by a second-order or fourth-order system (one or two resonance peaks in the electrical impedance magnitude). The required order is automatically determined by the fitting algorithm. If the tweeter’s rear volume is not sealed, please select Free Air or Vented Box.

Mechanical and Acoustical Calibration #

What is mechanical calibration?#

The FLSI determines loudspeaker parameters in the electrical domain from the measured voltage and current signals:

Loudspeaker Model in the electrical domain

To calculate the actual mechanical parameters such as moving mass and mechanical stiffness according to the full nonlinear loudspeaker model, the Bl(x=0) parameter must be known.

Automatic and Manual Mechanical Calibration#

It is strongly recommended to allow the FLSI to automatically determine this parameter using a laser. However, there may be situations where the laser does not have access. In such cases, the mechanical calibration can be performed by specifying Bl(x=0) or Bl_max on the property page. The latter refers to the maximum Bl value across the full displacement range, rather than just at the rest position (x=0).

If the calibration parameter must be entered manually, its value can be determined through a separate FLSI measurement of the transducer in free air. In this case, it is generally recommended to use the Bl_max calibration, as the rest position may shift during the large signal measurement. A significant change in position can, for example, often be observed when drivers are mounted in relatively small enclosures compared to their size (see Voice Coil Rest Position which explains this effect).

Added Mass Method#

Instead of the automatic mechanical calibration or specifying a Bl(x) or moving mass value manually, the Added Mass Method can be performed. Usually it is recommended to use a laser for the mechanical calibration. However, in some cases the added mass method is preferable, e.g.

if no laser is available
if the laser cannot be used due to bad optical conditions
validating the mechanical calibration using different measurement techniques
training and academic purposes

The added-mass method should be performed using the following procedure:

Create a new FLSI operation.
Apply an additional mass to the transducer’s cone, e.g. clay. Its mass should roughly be in the range of roughly 40-100 % of the estimated transducer’s moving mass.
Open the FLSI operation’s property page.
1. Configure the measurement (Speaker System Type, Limits, …)
2. In the Mechanical Calibration drop-down list, select Added Mass Method.
3. Press the button Create Operation for Added Mass Measurement. This creates a new operation which will be used to perform a small signal measurement with an additional mass applied to the transducer’s moving mass.
Run the newly created FLSI operation Measurement with Added Mass
Remove the moving mass.
Open the property page of the FLSI operation that you have created in step 1.
1. Press the button Import Measurement with Added Mass and select the FLSI operation Measurement with Added Mass of step 4.
2. Insert the precise value of the added mass in the Added Mass input field.
Run the operation.

Acoustical Calibration#

The acoustical calibration is optional. If an effective radiation area S_d(x=0) or a diameter is imported, more derived linear parameters are calculated and displayed in the linear parameter result table. The precise effective radiation area can be measured using SCN – Scanning Vibrometer. If the precise value cannot be measured by SCN and is not known, S_d(x=0) of circular drivers can be approximated by measuring the effective diameter d_d(x=0) as follows:

\[d_\mathrm{d}(x=0) \approx d_\mathrm{i} + \frac{(d_\mathrm{o} - d_\mathrm{i})}{2}\]

\[S_\mathrm{d}(x=0) = \pi \left(\frac{d}{2}\right)^2\]

In this case, allow slightly higher tolerances of the derived linear parameters that depend on S_d(x=0). Do not enter the nominal diameter of the driver such as 4” or 5.25”.

The calibration parameters are post-processing parameters that can be changed after the measurement.

Results #

Linear Parameters at x=0 #

Linear parameters are required as input parameters for traditional linear modeling. They are assumed to be constant and independent of the excitation — the dependence on state quantities (e.g. displacement x and temperature T_v) is neglected. .. In case the parameters depend on displacement, the linear parameters are defined at the rest position (x =0).

Linear Modeling is valid in the small-signal domain only. At higher amplitudes, there are systematic discrepancies between small-signal and large-signal parameters due to heating and the time variance of some transducer parameters.

The traditional loudspeaker design is based on a linear modeling comprising constant parameters only. This model is simple and explains the linear transfer behavior (e.g. amplitude and phase response) at low amplitudes. The linear model can be considered as an approximation of the expanded model valid in the small signal domain. The parameters of both models are closely related with each other. However, to explain the differences we have to distinguish between small signal parameters and large signal parameters at the rest position.

The table Linear Parameter at x=0 table displays the parameters of the lumped parameter model in the small signal domain and in the large signal domain at the voice coil rest position x=0.

If the Nonlinear Mode is activated on the FLSI’s property page, three columns are displayed:

LARGE+WARM (large signal domain + warm speaker)
- measured in the NONLINEAR MODE
- the peak voice coil displacement is high: |x|_peak = x_prot,
- the variation of the parameters is not negligible: C_min << 100 % and/or Bl_min <<100 %,
- the voice coil temperature might be increased (∆T_v > 0) due to heating.
LARGE+COLD (large signal domain + cold speaker)
- measured in the LINEAR and NONLINEAR MODE
- the peak voice coil displacement is high: |x|_peak = x_prot,
- the variation of the parameters is not negligible: C_min << 100 % and/or Bl_min <<100 %,
- the effect of heating is compensated while considering the cold voice coil resistance measured in the LINEAR MODE where ∆T_v = 0.
SMALL SIGNAL (small signal domain + cold speaker)
- measured in the LINEAR MODE
- the amplitude of the excitation signal is small,
- the displacement is small in comparison to the allowed maximum displacement: |x|_peak << x_prot,
- the variations of the nonlinear parameters are negligible: C_min ≈ 100 %, Bl_min ≈ 100 %,
- the increase of voice coil temperature is negligible: ∆T_v ≈ 0,
- the effects of the nonlinear, thermal and time-varying mechanisms are negligible and the transducer behaves almost linear.

The different columns of this table gives valuable information to investigate the time-variance of the lumped parameters. The time-variant suspension part parameters K_ms(t) and R_ms(t) (see Mechanical Resonator) and their derived parameters such as resonance frequency and Q-factors are of special interest in many cases. E.g., for the alignment of transducer and enclosure, it is not sufficient to only rely on the small signal parameters, because they can significant change in a short period over time.

In the :ref:`` section, the linear inductance impedance Z_l(f, x=0, i=0) was introduced as a frequency-dependent electrical lumped element. This is not suitable an export into simulation tools (such as SIM). Therefore LR-3 inductance model parameters are displayed here to feed into these tools.

Note

The LR-3 inductance model parameters are valuable as input for simulation tools. However, they are not physically interpretable and cannot be directly compared between different transducers. For this reason, it is recommended to compare the frequency-dependent impedance curves \(Z_\mathrm{L}(f)\) instead, which provide a physically meaningful basis for comparison.

Linear Transfer Functions #

Electrical Impedance#

The complex electrical impedance is the transfer function between voltage and current

\[Z_\mathrm{e}(f) = \frac{U(f)}{I(f)}.\]

This transfer function can both be measured and calculated using the identified linear parameters. The FLSI’s electrical impedance result window displays the magnitude and the phase (right-click in the window and activate Show All Curves) of the following impedances:

Magnitude / Phase (measured)
- measured impedance in the small signal domain (LINEAR MODE)
Magnitude / Phase (fitted)
- fitted impedance that is calculated based on the parameters identified in the small signal domain (LINEAR MODE)
- includes mechanical creep
Magnitude / Phase (Thiele-Small simplification)
- fitted impedance that is calculated based on the parameters identified in the small signal domain (LINEAR MODE)
- does not include mechanical creep
- corresponds to the Small Signal parameters displayed in Linear Parameters at x=0
Magnitude / Phase (large signal at x=0)
- fitted impedance that is calculated based on the parameters identified in the small signal domain (NONLINEAR MODE)
- includes time-variant parameters and mechanical creep

Lossy Inductance#

The coil is surrounded by permeable and conductive materials, which cause

iron losses due to eddy currents, the skin effect, and hysteresis, as well as
conductive losses due to shorting materials (see Self-Inductance).

The frequency-dependent lossy inductance in the small signal domain, measured in LINEAR MODE at at \(x = 0, i = 0\), is represented in two ways:

Magnitude and Phase of the Impedance of the Inductance:

\[\underline{Z_\mathrm{L}}(f) = |Z_\mathrm{L}(f)| e^{i\phi_\mathrm{L}(f)}\]
Lossy Inductance and Resistance:

\[\underline{Z_\mathrm{L}}(f) = 2\pi f j L(f) + R_\mathrm{L}(f).\]

This representation can be easier to interpret than the magnitude and phase. For instance, the losses that are transformed into heat can be seen directly for each frequency and compared to the DC resistance R_e.

Mechanical Stiffness Km(f,x=0)#

This window shows the frequency-dependent small signal stiffness K_m(f, x) in the small signal domain, measured in the LINEAR MODE. See Mechanical Resonator for further explanation.

If no frequency-dependency was detected, K_m is constant over frequency. It is only displayed for 2nd order mechanical systems (free air and closed speaker type).

In case a free air system is measured, the stiffness K_ms is displayed. The cause of the frequency-dependency is the mechanical creep effect. In case a closed box system is measured, the total stiffness K_mt is displayed. This comprises the transducer’s suspension stiffness K_ms and the stiffness of the enclosed air K_mb. In this case, the stiffness frequency dependency can either be caused by creep, air leakage, or both.

Displacement Frequency Response#

The linear transfer function between the voice coil displacement X(f) and the excitation voltage U(f)

\[H_{\textrm{x}}(f) = \frac{X(f)}{U(f)}\]

can be measured with a laser or calculated using the identified linear parameters like resonance frequency, Q-factors and the force factor. The two transfer functions should be almost identical. If not, the laser measurement might have been disturbed or the model was not identified with sufficient accuracy.

The transfer functions can be calculated in both the small signal (LINEAR MODE) and the large signal (NONLINEAR MODE) measurement. Calculating the laser displacement transfer function in the small signal domain is relatively simple, because the laser and voltage signals can be divided in the frequency-domain without any further processing. However in the large signal domain, the processing is more complex because linear transfer functions are only defined for linear signals. Hence, the nonlinear distortion which was identified in the parameter identification algorithm is removed from the measured nonlinear laser signal before calculating the transfer function.

The peak displacement frequency response is calculated by multiplying this transfer function by the crest factor of a sinus tone \(\sqrt 2\).

\[H_{\textrm{x,peak}}(f) = \frac{X_\textrm{peak}(f)}{U_\textrm{RMS}(f)} = \sqrt 2 H_{\textrm{x}}(f)\]

This peak displacement frequency response is highly valuable for estimating how much peak displacement can be reached with a sinus tone of a certain RMS voltage. The peak displacement produced by a 1 V_RMS sinus tone can be directly read from this curve. Note that this estimation is purely linear and does not consider nonlinear distortion.

The following displacement frequency response magnitudes are displayed:

Laser (small signal)
- measured \(|H_{\textrm{x,peak}}(f)|\) in the small signal domain (LINEAR MODE)
Fitted (small signal)
- fitted \(|H_{\textrm{x,peak}}(f)|\) that is calculated based on the parameters identified in the small signal domain (LINEAR MODE)
- includes mechanical creep
Thiele-Small (small signal)
- fitted \(|H_{\textrm{x,peak}}(f)|\) that is calculated based on the parameters identified in the small signal domain (LINEAR MODE)
- does not include mechanical creep
- corresponds to the Small Signal parameters displayed in Linear Parameters at x=0
Laser (large signal)
- linearized large signal displacement frequency response with \(X_\textrm{nonlinear}(f)\) which is the identified nonlinear part of the displacement
\[|H_{\textrm{x,peak,lin}}(f)| = \sqrt 2 \frac{X_\textrm{laser}(f) - X_\textrm{nonlinear}(f)}{U_\textrm{RMS}(f)}\]
- includes time-variant parameters and mechanical creep
Fitted (large signal)
- fitted \(|H_{\textrm{x,peak}}(f)|\) that is calculated based on the parameters identified in the large signal domain (NONLINEAR MODE)
- includes mechanical creep

Large Signal Parameters (Curves)#

Check Relevance of Nonlinear Parameters#

FLSI identifies parameters by minimizing the error between measured and simulated states (primarily the electric current). If a parameter — such as Bl(x) or K_ms(x) — has little or no influence on the states, the sensitivity of parameter fitting is low. This means that:

The parameter has negligible influence on the loudspeaker’s behavior and performance.
The parameter should not be interpreted further — it generally does not contain relevant information.
The shape of the nonlinear parameter may be imprecise because no useful information could be extracted from the underlying states.

To check whether a parameter is relevant for the device’s behavior, review:

Distortion Components in Large Signal Characteristics (Single Values): If the distortion component’s value is very small (less than 1 %), the parameter is usually negligible. In this case, the distortion component value is displayed in light gray.
Displacement Limits: If the displacement limit is greater than the maximum displacement reached during the measurement, the parameter may be irrelevant. Confirm by also checking the distortion components.

Note that a parameter that is irrelevant in one measurement can be important in another with different conditions:

Increasing the excitation voltage can generate more distortion, making a parameter more significant.
Mounting the transducer in an enclosure can make parameters important that were negligible in a free-air measurement, due to different operating conditions (e.g., stiffness or current).

Voice Coil Rest Position#

In all nonlinear parameter curves that depend on displacement x, a solid black vertical line is shown. This line represents the instantaneous rest position of the voice coil.

If a laser is used, the measured rest position — the position the coil would return to if the stimulus were turned off — is displayed as a dotted vertical line.

A deviation between the dotted and solid lines indicates that one of the following conditions may be present:

The voice coil rest position has shifted during the large-signal measurement.

This can occur if the suspension changes its geometry during the test. Such a shift is usually an undesired defect.
An unmodeled DC force is present.

This effect is sometimes observed in small, closed enclosures: Closed boxes are never perfectly sealed; they must include a barometric vent to equalize pressure between the enclosure’s interior and the surrounding air. In such cases, a DC force — often caused by a nonlinearity like the reluctance force — acts not on the total stiffness (K_mt), but primarily on the mechanical stiffness (K_ms). As a result, the DC force acting only on K_ms causes a larger DC displacement than if it acted on K_mt (the stiffness determined by the parameter identification). In this case, the parameter identification algorithm may underestimate the actual DC displacement which results in a shifted instantaneous rest position.
The laser or DUT is not in a fixed position.

This is a measurement setup error and should be corrected.

Force Factor Bl(x)#

The force factor \(Bl(x)\), also called electrodynamic coupling factor or electromechanical transduction factor, is defined by the integral of the magnetic flux density B over voice coil length l.

The solid gray curve \(-Bl(x)\) represents the force factor mirrored at the rest position. This visualization helps to identify any asymmetries in the suspension.

When interpreting this curve, check if the parameter is relevant for this loudspeaker’s behavior (see Check Relevance of Nonlinear Parameters).

Mechanical Stiffness Km(x)#

The stiffness \(K_\mathrm{ms}(x)\) characterizes the mechanical properties of the suspension and is the inverse of the compliance \(C_\mathrm{ms}(x)\). The nonlinear stiffness is defined as the secant stiffness of the lossless restoring force–deflection curve characteristic \(F(x)\):

\[K_\mathrm{ms}(x) = \frac{F(x) - F(x_0)}{x - x_0}\]

where \(x_0 = 0\) denotes the voice-coil rest position.

The solid gray curve \(K_\mathrm{ms}(-x)\) represents the stiffness mirrored at the rest position. This visualization helps to identify any asymmetries in the suspension.

If the speaker’s enclosure is closed, the chart displays the total stiffness of the system:

\[K_\mathrm{mt}(x) = K_\mathrm{ms}(x) + K_\mathrm{mb}(x)\]

In order to display the driver stiffness \(K_\mathrm{ms}(x)\) and the stiffness of the enclosed air \(K_\mathrm{mb}(x)\), it is necessary to specify \(S_d(x=0)\) and the box volume \(V_\mathrm{b}\) on the property page (see Acoustical Calibration). The parameters \(S_d(x=0)\) and \(V_\mathrm{b}\) can be changed after the measurement is complete. The enclosure type must be set before the measurement, though.

The small-signal stiffness, measured in the LINEAR MODE, is displayed alongside the nonlinear stiffness, measured at high amplitude during the NONLINEAR MODE. This allows for a direct comparison of how the mechanical stiffness changes at the rest position across different levels.

When interpreting this curve, check if the parameter is relevant for this loudspeaker’s behavior (see Check Relevance of Nonlinear Parameters).

Note

FLSI also calculates the incremental (small-signal) stiffness, which can be accessed by right-clicking in the \(K_\mathrm{m}(x)\) window and selecting Show All Curves. The incremental stiffness is defined as the local derivative of the restoring force with respect to displacement:

\[K_\mathrm{incr}(x_k) = \left.\frac{\mathrm{d}F(x)}{\mathrm{d}x}\right|_{x=x_k} ,\]

Note

The nonlinear compliance curves \(C_\mathrm{ms}(x)\) and \(K_\mathrm{mt}(x)\) can be accessed by right-clicking in the \(K_\mathrm{m}(x)\) window and selecting Show All Curves. Viewing these curves can be useful for understanding the protection system, as the protection limit \(C_\mathrm{min}\) refers to the nonlinear compliance.

Lossy Inductance (x)#

As described in Self-Inductance, the voice coil’s self-inductance depends on frequency, displacement, and current:

\[\underline{Z}_\mathrm{L}(f,x,i) = 2\pi f j L(f,x,i) + R_\mathrm{L}(f,x,i).\]

Accurately modeling all electro-dynamic effects for real audio signals over a wide bandwidth requires a complex nonlinear dynamic model, because frequency, displacement, and current are interdependent in a non-trivial way. However, the results of such a model cannot easily be visualized for intuitive interpretation.

To make the displacement-dependency of inductance interpretable, the model is simplified to a set of blocked impedances:

\[\underline{Z}_\mathrm{L,blocked}(f,x_\mathrm{DC},i\approx 0),\]

considering only the DC part of the total displacement which comprises a DC and an AC part:

\[x = x_\mathrm{DC} + x_\mathrm{AC}.\]

The electric current flow (current dependency is covered in Flux Modulation) is assumed to be negligible for displacement-dependency investigations.

The blocked impedance representation is comparable to the results of a semi-static point-by-point measurement (see IEC 62458 or Measuring the Nonlinear, Lossy, Frequency-dependent Voice Coil Inductance), where the voice coil is rigidly fixed at different positions \(x_\mathrm{DC}\), and small-signal measurements are performed to obtain the electrical impedance curves at these positions.

Note

For simplicity, FLSI’s user interface omits the index \(DC\) and does not explicitly indicate \(i\approx 0\):

\[\underline{Z}_\mathrm{L}(f,x) \equiv \underline{Z}_\mathrm{L,blocked}(f,x_\mathrm{DC},i\approx 0)\]

Note

In Klippel’s previous large-signal identification software (LSI2 and LSI3), inductance variation was displayed using a pure inductance \(L_\mathrm{e}(x)\) (also see Appendices, explaining differences between LSI and FLSI).

A more insightful representation is to evaluate the impedance magnitude of the inductance \(|\underline{Z}_\mathrm{L}(f,x)|,\) calculated at selected frequencies \(f\). This directly shows how impedance changes with displacement and frequency, and relates directly to the amplitude modulation produced by inductance nonlinearity. For example, a large variation in impedance magnitude produces greater amplitude modulation than a small variation.

The pure inductance \(L(f,x)\) shows only part of the distortion mechanism. Inductance nonlinearity causes both amplitude and phase modulation — with the phase component having much less perceptual impact on audible intermodulation distortion (further reading: Psychoacoustics - Facts and Models).

In most drivers, inductance exhibits a strongly asymmetric characteristic at low frequencies: When the voice coil moves toward the back plate, inductance typically increases because the magnetic field generated by the coil encounters lower magnetic reluctance due to the high permeability of iron. This property can be used to verify loudspeaker polarity and interpret the excursion direction in diagrams of nonlinear parameters.

The shape of the nonlinearity (symmetric vs. asymmetric) also provides insight into the type of nonlinear distortion:

Asymmetric behavior typically produces second-order distortion intermodulation distortion.
Symmetric behavior primarily produces third-order intermodulation distortion.

Note

\(|\underline{Z}_\mathrm{L}(f,x)|\) should not be interpreted without considering the voice coil’s DC resistance \(R_\mathrm{e}\), which is also shown in the chart. If \(|\underline{Z}_\mathrm{L}(f,x)|\) is significantly smaller than \(R_\mathrm{e}\) — as is often the case in micro-speakers and woofers at low frequencies — the distortion caused by self-inductance is usually negligible.

Another useful representation of lossy inductance, showing its dependency on frequency and voice coil position, is obtained by calculating the ratio between the AC flux linkage \(\Lambda_\mathrm{AC}(f,x)\) (Fourier-transformed flux linkage \(\lambda\mathrm{AC}(t,x)\)) and the corresponding electric current \(I(f)\):

\[L_\mathrm{l}(f,x) \equiv \left|\frac{\underline\Lambda_\mathrm{AC}(f,x)}{\underline I(f)}\right| = \frac{\left|Z_\mathrm{L}(f,x)\right|}{2\pi f}\]

Here, the lossy inductance \(L_\mathrm{l}(f,x)\) (in henries) can be interpreted as an inductance derived from the impedance magnitude — it ignores the phase of \(\underline{Z}_\mathrm{L}\) and treats all impedance as purely inductive.

This quantity is displayed as a color map (contour plot) showing the AC flux linkage distribution for different voice coil positions (vertical axis) and frequencies (horizontal axis).

Note

While the impedance-magnitude chart \(|\underline{Z}_\mathrm{L}(f,x)|\) mainly reveals the symptoms of nonlinear inductive effects, the \(L_\mathrm{l}(f,x)\) representation is more directly related to the underlying physical phenomena in the motor — particularly, the variation of magnetic flux linkage with coil displacement and signal frequency.

When interpreting this curve, check if the parameter is relevant for this loudspeaker’s behavior (see Check Relevance of Nonlinear Parameters).

Flux Modulation#

The total magnetic flux “seen” by the voice coil in the iron parts of the pole plates and pole piece consists of two components at the voice coil rest position:

a DC component \(\lambda_\mathrm{0}\), produced by the permanent magnet
the flux linkage \(\lambda(i_\mathrm{ac})\), produced by the electric AC current in the voice coil.

If the AC voice-coil current significantly alters the total magnetic flux and the permeability of the iron in the skin region, the effective permeability, and therefore the inductance, becomes modulated. However, if the iron is already highly saturated by the static flux from the permanent magnet, this variation is typically small.

In FLSI, the flux modulation is represented as the relative effective flux linkage

\[\frac{\lambda(i_\mathrm{ac})}{\lambda(i=0)},\]

measured at the voice coil’s rest position \(x=0\).

B-H curve of iron explaining flux modulation

Flux modulation primarily contributes to harmonic distortion at frequencies above the loudspeaker’s resonance frequency.

When interpreting this curve, check if the parameter is relevant for this loudspeaker’s behavior (see Check Relevance of Nonlinear Parameters).

Note

The flux modulation identified in FLSI is valid for AC signals at typical audio frequencies. If the skin effect’s penetration depth, \(\delta_\mathrm{skin}\) is significantly smaller than the pole piece diameter, \(d_\mathrm{pole}\) — which is typically the case already at low frequencies (20 Hz) — the flux modulation affects only the voice coil’s self-inductance and has no significant impact on \(Bl\):

\[Bl(i) \approx \text{constant with respect to } i, \quad \text{when } \delta_{\mathrm{skin}} \ll d_{\mathrm{pole}}\]

Flux Density with 20 Hz sinusoidal excitation, visualizing the skin effect (FEMM)

Note that the effective flux modulation is not a parameter in the strict sense, as the nonlinear curve shape depends on the particular excitation signal, its bandwidth, frequency shaping, and other signal characteristics.

Mechanical Resistance Rms(v)#

The dependency of the mechanical resistance \(R_\mathrm{ms}(v)\) on the voice coil velocity v is a dominant nonlinearity in micro-speakers and other transducers that have relatively small motor systems, including smaller woofers. This nonlinearity can be usually neglected in larger woofers.

The nonlinear variation of \(R_\mathrm{ms}(v)\) is not due to the mechanical motion of the diaphragm or other mechanical elements, but instead arises from the nonlinear air-flow resistance in small structures, typically within the transducer’s motor. This effect can be easily verified by performing a measurement in vacuum, where the nonlinearity disappears. In FLSI, this acoustic parameter is reflected into the mechanical domain via \((Bl)^2\), so that the resulting nonlinearity becomes comparable to \(R_\mathrm{ms}(v \approx 0)\), which represents the small-signal mechanical losses.

The nonlinearity of \(R_\mathrm{ms}(v)\) leads to a significant increase in mechanical damping near the resonance frequency, resulting in nonlinear amplitude compression of the fundamental and the generation of substantial harmonic and intermodulation distortion.

When interpreting this curve, check if the parameter is relevant for this loudspeaker’s behavior (see Check Relevance of Nonlinear Parameters).

Symmetry Ranges#

Bl(x) Symmetry Range#

The asymmetry of the Bl(x) curve can be represented by a single value, as defined in Nonlinear Parameters. It can also be calculated for different voice coil offsets, \(x_\mathrm{off}\), and amplitudes, \(x_\mathrm{ac}\), resulting in \(a_{\mathrm{Bl}} \left(x_\mathrm{ac}, x_\mathrm{off} \right)\).

An asymmetry is considered negligible if \(a_{\mathrm{Bl}} \left(x_\mathrm{ac}, x_\mathrm{off} \right) < 5\ %\). A so-called Bl(x) symmetry region is displayed as a gray area in the Bl(x) symmetry window, plotted against x_off and x_ac.

When interpreting this curve, check if the parameter is relevant for this loudspeaker’s behavior (see Check Relevance of Nonlinear Parameters).

Km(x) Symmetry Range#

The K_ms(x) or K_mt(x) (in case of closed box speaker type) symmetry range represents the region where the stiffness variation remains below 5%, as function of displacement amplitude (x-axis) and voice coil offset (y-axis). It is calculated in a similar way as the Bl(x) symmetry range.

Within the desired operating range, the rest position — indicated by the thick black curve — should lie entirely within the symmetry range (gray area). In this case, K_ms(x) (or K_mt(x)) asymmetries can be considered negligible.

A typical indicator of a geometrical asymmetry in the spider (e.g., cup shape) or surround (e.g., half-wave profile) is when the symmetry region runs parallel to the x-axis but never intersects it. In some cases, improved alignment of the suspension components may help. However, the root causes of asymmetry typically need to be identified by separating the spider and surround and using finite element analysis (FEA).

When interpreting this curve, check if the parameter is relevant for this loudspeaker’s behavior (see Check Relevance of Nonlinear Parameters).

Note

The symmetry range, displayed as gray area, was was changed from 5% in LSI to 20% in FLSI. We have learned that a symmetry range limited at 5% was too critical, leading to an overestimation of the negative effects.

Bl(x) and Km(x) Symmetry Points#

The red lines in the Bl(x) and K_m(x) symmetry range charts represent their symmetry points. Ideally, the symmetry points coincide with the x-axis.

The symmetry point in the nonlinear Bl(x) curve is where a negative and positive displacement x will produce the same force factor

\[Bl(x + x_\mathrm{sym,bl}) \overset{!}{=} Bl(-x + x_\mathrm{sym,bl})\]

with \(x_\mathrm{sym,bl}\) being the voice coil shift that symmetrizes Bl(x) for the amplitude x.

The same rule applies to the K_ms(x) or K_mt(x) symmetry point:

\[K_\mathrm{m}(x + x_\mathrm{sym,k}) \overset{!}{=} K_\mathrm{m}(-x + x_\mathrm{sym,k}).\]

Large Signal Characteristics (Single Values)#

Distortion Components#

See Distortion Components.

Displacement Limits#

FLSI calculates parameter-based displacement limits according to standard IEC 62458 or based on manual thresholds.

Force Factor:

The maximum displacement \(x_{\mathrm{Bl}}\) is limited by excessive nonlinear Bl(x) distortion. It is defined as the smallest displacement for which the relative force factor

\[\frac{Bl(x)}{Bl(0)}\]

reaches a defined threshold \(Bl_{\mathrm{min}}\) (default: 82%):

\[x_{\mathrm{Bl}} = \min \left\{ |x| \; : \; \left| \frac{Bl(x)}{Bl(0)} \right| = Bl_{\mathrm{min}} \right\}.\]

Stiffness/Compliance::

The maximum displacement \(x_{\mathrm{C}}\) is limited by excessive nonlinear stiffness/compliance distortion. It is defined as the smallest displacement for which the relative mechanical compliance (which is the reciprocal of stiffness)

\[\frac{C_\mathrm{m}(x)}{C_\mathrm{m}(0)} = \frac{K_\mathrm{m}(0)}{K_\mathrm{m}(x)}\]

reaches a defined threshold \(C_{\mathrm{min}}\) (default: 75%):

\[x_{\mathrm{C}} = \min \left\{ |x| \; : \; \left| \frac{K_\mathrm{m}(0)}{K_\mathrm{m}(x)} \right| = C_{\mathrm{min}} \right\}.\]

For free air systems, \(K_\mathrm{m}\) corresponds to \(K_\mathrm{ms}\). For speaker systems comprising a closed back chamber, \(K_\mathrm{m} \equiv K_\mathrm{mt}\) (total stiffness comprising the mechanical suspension \(K_\mathrm{ms}\) and the air spring stiffness \(K_\mathrm{mb}\)).

Inductance:

The amplitude modulation caused by the nonlinear self-inductance is directly related to the variation of the magnitude of the blocked electrical impedance with displacement \(x\). This impedance, comprising only the DC resistance and the voice coil (not considering back-EMF), is defined as:

\[Z_{\mathrm{e}}(f,x) = R_{\mathrm{e}} + Z_{\mathrm{L}}(f,x, i \approx 0)\]

The displacement limit \(x_{\mathrm{L}}\) is defined as the smallest displacement for which the relative impedance magnitude

\[\frac{|Z_{\mathrm{e}}(f_2,x)|}{|Z_{\mathrm{e}}(f_2,0)|}\]

varies by a defined threshold \(Z_{\mathrm{max}}\) (default: 10%):

\[x_{\mathrm{L}} = \min \left\{ |x| \; : \; \left| \frac{|Z_\mathrm{e}(f_2,x)|}{|Z_\mathrm{e}(f_2,0)|} - 1 \right| = Z_{\mathrm{max}} \right\}\]

\(f_{2}\) is the frequency of the modulated voice tone and set to \(f_{2} = 8.5 f_{\mathrm{res}}\).

For free air and vented box measurements, \(f_{\mathrm{res}}\) corresponds to the transducer’s resonance frequency \(f_{\mathrm{s}}\). In other complete system measurements (i.e., closed box), \(f_{\mathrm{res}} = f_{\mathrm{c}}\).

Note

\(x_{\mathrm{L}}\) can be read from the inductance nonlinearity chart Lossy Inductance (x): It corresponds with the minimum displacement, where the inductance impedance magnitude curve at \(f_{2}\), plus the small signal DC resistance, (highlighted by a thick curve) varies by \(\pm Z_{\mathrm{max}}\).

Doppler:

The maximum displacement \(x_\mathrm{D}\) is defined as the point where the Doppler Effect introduces excessive phase modulation. It can be estimated analytically using the expression

\[x_{\mathrm{D}} = \frac{770 \cdot d_{\mathrm{2}}}{f_{\mathrm{2}}}\]

This equation, presented by Beers and Belar, determines the peak displacement \(x_\mathrm{D}\) (in mm), where

\(d_\mathrm{2}\) is the second-order modulation distortion in percent, and
\(f_\mathrm{2}\) is the frequency of the modulated voice tone.

Following their approach, we set

\[f_{2} = 8.5 f_{\mathrm{res}}\]

and assume a distortion threshold of \(d_\mathrm{2} = 10\%\). Substituting these values yields the Doppler displacement limit

\[x_{\mathrm{D}} = \frac{90.5 \cdot d_\mathrm{2}}{f_{\mathrm{res}}}\]

where \(x_\mathrm{D}\) is in millimeters and \(f_\mathrm{res}\) is in Hertz.

Thermal Parameters#

See the Thermal model.

Nonlinear Parameters#

Maximum Force Factor Bl max:

Bl(x) may reach its maximum at a displacement other than the rest position (x=0) if the curve is asymmetric. Bl_max denotes the maximum Bl value over the entire modeled displacement range.

\[Bl_{\mathrm{max}} = \max\limits_{|x| \leq x_\mathrm{prot}} Bl(x)\]

Maximum Modeled Displacement xprot:

x_prot is the maximum absolute displacement predicted by the nonlinear loudspeaker model:

\[x_\mathrm{prot} = \max\limits_{0\leq t\leq T} |x_\mathrm{modeled}|.\]

This value may slightly differ from the maximum displacement measured by the laser due to small modeling and measurement inaccuracies. The complete modeled and measured displacement signals can be compared in the Displacement Waveform Chart.

Maximum Port Velocity:

The maximum of the modeled mean spatial particle velocity \(v_{\mathrm{p,modeled}}\) in the port

\[v_{\mathrm{p,peak}} = \max\limits_{0\leq t\leq T} |v_{\mathrm{p,modeled}}|\]

can be calculated based on the full nonlinear lumped parameter model and precise values of the effective radiation area of the transducer and port, specified on the FLSI’s property page.

Bl(x) Symmetry point x sym,bl:

This parameter is a single-value representation of the symmetry point curve displayed in the Bl(x) Symmetry Range window at the displacement x_prot.

The symmetry point is specified in the standard IEC 62458.

Km(x) Symmetry point x sym,k:

This parameter is a single-value representation of the symmetry point curve displayed in the Km(x) Symmetry Range window at the displacement x_prot.

It is calculated in a similar way as the Bl(x) symmetry point.

Bl(x) Asymmetry:

The Bl(x) asymmetry can be assessed by a single value, as defined in Nonlinear Parameters.

\[a_\mathrm{bl}\left( x_{\mathrm{prot}} \right) = 2 \cdot \frac{Bl\left( - x_{\mathrm{prot}} \right) - Bl\left(x_{\mathrm{prot}} \right)}{Bl\left(-x_{\mathrm{prot}} \right) + Bl\left(x_{\mathrm{prot}} \right)}100\%.\]

This asymmetry parameter is specified in the standard IEC 60268-22. A positive asymmetry value usually corresponds to a negative symmetry point.

Stiffness Asymmetry:

The K_m stiffness asymmetry can be assessed by a single value

\[a_\mathrm{k}\left( x_{\mathrm{prot}} \right) = 2 \cdot \frac{K_{\mathrm{m}}\left( - x_{\mathrm{prot}} \right) - K_{\mathrm{m}}\left( x_{\mathrm{prot}} \right)}{K_{\mathrm{m}}\left( - x_{\mathrm{prot}} \right) + K_{\mathrm{m}}\left( x_{\mathrm{prot}} \right)}100\%,\]

using the stiffness at the negative and positive maximum displacement ± x_prot of the measured K_m(x)-curve. The sign of a_k corresponds to the direction of the DC displacement generated dynamically by the nonlinear rectification process of the displacement. It has the same sign like the stiffness symmetry point.

The stiffness asymmetry parameter is specified in the standard IEC 60268-22.

Port Asymmetry:

This value is displayed only for vented box systems. The asymmetry of the nonlinear port resistance \(R_\mathrm{ap}(v_\mathrm{p})\) (see the lumped parameter model of the ported system in section Speaker System Modeling) can be assessed using the following equation:

\[a_\mathrm{r,port}\left( v_{\mathrm{p,peak}} \right) = 2 \cdot \frac{R_{\mathrm{ap}}\left(v_{\mathrm{p,peak}} \right) - R_{\mathrm{ap}}\left(-v_{\mathrm{p,peak}} \right)}{R_{\mathrm{ap}}\left( - v_{\mathrm{p,peak}} \right) + R_{\mathrm{ap}}\left( v_{\mathrm{p,peak}} \right)}100\%,\]

This calculates the asymmetry in the nonlinear port resistance at the maximum peak velocities observed during the measurement. At the negative peak velocity \(-v_{\mathrm{p,peak}}\), air is sucked into the enclosure. At the positive peak velocity \(+v_{\mathrm{p,peak}}\), air is pushed out of the enclosure into the ambience. A negative sign of \(a_\mathrm{r,port}\) means that the resistance to push air outside the enclosure into the ambience is lower than to suck air inside, and vice versa.

Any asymmetry of the port resistance creates a static pressure difference between the atmospheric static pressure outside the loudspeaker enclosure and its inside. This static pressure difference creates a DC displacement of the voice coil that has to be considered in the system design to ensure stability and robustness.

Note

The sign of the voice coil DC displacement caused by the port asymmetry corresponds with the sign of the asymmetry value.

Minimal Force Factor Ratio Bl min:

The minimal force factor ratio

\[Bl_{\mathrm{min}} = \min\limits_{|x| \leq x_\mathrm{prot}}\left( \frac{Bl\left( x \right)}{Bl\left( 0 \right)} \right)\]

is a single value representation of the nonlinear force factor Bl(x), neglecting its complicated curve shape. It plays a crucial role in determining x_prot and in providing mechanical protection for the driver during the measurement.

Minimal Compliance Ratio C min:

The minimal compliance ratio

\[C_{\mathrm{min}} = \min\limits_{|x| \leq x_\mathrm{prot}}\left( \frac{C_{\mathrm{m}}\left( x \right)}{C_{\mathrm{m}}\left( 0 \right)} \right) = \min\limits_{|x| \leq x_\mathrm{prot}}\left( \frac{K_{\mathrm{m}}\left( 0 \right)}{K_{\mathrm{m}}\left( x \right)} \right),\]

is a single value representation of the nonlinear compliance, neglecting its complicated curve shape. It plays a crucial role in determining x_prot and in providing mechanical protection for the driver during the measurement.

Identification Accuracy#

General#

The following sections explain the metrics used to evaluate the accuracy of the identified nonlinear parameter model in both the small and large signal domains. Error values below 5% are typically considered good. If any error metrics exceed predefined thresholds, the FLSI will display warning or error messages.

Fitting Error (LINEAR MODE)#

The impedance fitting error of the small signal measurement (LINEAR MODE) is expressed as the normalized mean average error (NMAE)

\[E_\mathrm{lin} = \sqrt{\frac{\sum\limits_{n = 0}^{N - 1} \vert Z_{\text{fit}}(f_{n}) - Z_{\text{meas}}(f_{n}) \vert}{\sum\limits_{n = 0}^{N - 1} \vert Z_{\text{meas}}(f_{n}) \vert}}\]

between the measured and the fitted electrical impedance at all excited frequencies \(f_n\).

Displacement Fitting Error (LINEAR MODE)#

The error between the fitted voice coil displacement and the laser-measured displacement of the small signal measurement (LINEAR MODE) is expressed as the normalized root mean squared error (NRMSE)

\[E_\mathrm{x,lin} = \sqrt{\frac{\sum\limits_{n = 0}^{N - 1} \vert H_{\text{x,fit}}(f_{n}) - H_{\text{x,meas}}(f_{n}) \vert^{2}}{\sum\limits_{n = 0}^{N - 1} \vert H_{\text{x,meas}}(f_{n}) \vert^{2}}}\]

with the transfer function between displacement and voltage which is closely related to the displacement frequency response

\[H_\mathrm{x}(f_n) = \frac{X(f_n)}{U(f_n)}.\]

\(H_\mathrm{x}(f_n)\) is evaluated solely at low frequencies, typically below approximately four times the DUT’s resonance frequencies, because the laser signal becomes noisy at higher frequencies.

This error metric is highly valuable to evaluate the accuracy of the identified Bl(x=0), under the assumption that

the laser is correctly calibrated,
the optical conditions are good, and
no severe rocking modes occur.

Note that if an inaccurate Bl(x=0) is imported, this error will increase.

Fitting Error (NONLINEAR MODE)#

The fitting error of the large signal measurement is calculated using the NRMSE between the fitted and the measured electrical current:

\[E_\mathrm{nonlin} = \sqrt{\frac{\sum\limits_{n = 0}^{N - 1} \vert i_{\text{fit}}(t_{n}) - i_{\text{meas}}(t_{n}) \vert^2}{\sum\limits_{n = 0}^{N - 1} \vert i_{\text{meas}}(t_{n}) \vert^2}}\]

See High Fitting Error (LINEAR MODE / NONLINEAR MODE) which explains what do if this error is too high.

Displacement Fitting Error (NONLINEAR MODE)#

The error between the fitted voice coil displacement and the laser-measured displacement of the large signal measurement (NONLINEAR MODE) is expressed as the NRMSE

\[E_\mathrm{x,lin} = \sqrt{\frac{\sum\limits_{n = 0}^{N - 1} \vert X_{\text{fit}}(f_{n}) - X_{\text{meas}}(f_{n}) \vert^{2}}{\sum\limits_{n = 0}^{N - 1} \vert X_{\text{meas}}(f_{n}) \vert^{2}}}.\]

This error metric is closely related to Displacement Fitting Error (LINEAR MODE)

States #

Waveforms and Spectra#

The voltage and current waveforms show the electrical signals measured by the KA3’s sensors, while the sound pressure waveform represents the microphone signal. All signals are displayed in the time domain.

The spectra show how the signals are distributed across frequencies in the frequency domain. The following curves are available:

Fundamental
The multi-tone stimulus used by the FLSI consists of many different frequencies. This curve displays the measured signal at those frequencies. You can right-click in the chart and press Mark Data Points, or use the A key to highlight them. Note that a single low-frequency tone is applied to measure the voice coil’s DC resistance.
Noise
A short noise floor measurement is performed at the start of the LINEAR MODE. Note that the noise floor measurement in the LINEAR MODE is shorter than in the NONLINEAR MODE. As a result, the noise floor spectrum has a lower frequency resolution than the Distortion+Noise curve.
Distortion+Noise
In an ideal, linear, noise-free system, no signal energy exists between the fundamental frequencies. However, a loudspeaker is a nonlinear system that generates new frequency components as combinations of the fundamental frequencies.

Displacement Waveform and Spectrum#

The laser-measured displacement is displayed as both a waveform and a spectrum. In addition to the measured signal, the displacement calculated from the nonlinear lumped loudspeaker model is displayed. The deviation between the modeled and measured displacement is typically very low. However, if an inaccurate Bl(x=0) is imported through the property page, the deviation will be significant because the mechanical calibration will be incorrect.

You can right-click in the chart and select Show all Curves to display the displacement calculated using a linear lumped parameter model. In a highly nonlinear loudspeaker, the deviation between the peak displacement of the nonlinear and linear models can exceed 50%. Comparing the two modeled displacement signals indicates the error introduced by linear simulations, such as those used in linear speaker protection systems.

Distortion Components#

The sound pressure output for the multi-tone signal used in the NONLINEAR MODE can be estimated through the identified nonlinear model. This numerical simulation also allows for the evaluation of the nonlinear distortion generated by any nonlinearity present in the model, providing valuable insights into how each nonlinearity contributes to the overall distortion.

This capability is highly useful for identifying the root cause of the dominant nonlinear distortion. It can guide efforts to optimize the relevant components of the loudspeaker assembly.

Furthermore, it highlights which nonlinearities have a negligible impact on the speaker’s performance. For example, distortion components below 1% are typically considered minor and generally don’t require significant attention.

Additionally, the simulation can evaluate the impact of various factors–such as stimulus level, linear transducer parameters, and enclosure type (e.g., closed, vented, or passive radiator)–on the total distortion.

The FLSI displays the distortion components in two ways:

SPL Spectrum with Distortion Components: This window displays the spectrum for each individual distortion component, offering insights into the distortion characteristics of each nonlinearity. It shows, for example, in which frequency range a nonlinearity is most dominant. These distortions arise from the various combinations of fundamental frequencies. The distortion components include both harmonic and intermodulation distortion.
Relative Nonlinear Distortion Components: These single values displayed in a table in the Large Signal Parameters window show the overall distortion resulting from all the nonlinearities.

The relative distortion components are calculated using the ratio between the RMS value of each distortion component, denoted as \(p_{\mathrm{RMS},i}\), and the total sound pressure \(p_{\mathrm{RMS,total}}\):

\[D_{i} = \frac{p_{\mathrm{RMS},i}}{p_{\mathrm{RMS,total}}}.\]

The RMS value of a signal is calculated using

\[p_\textrm{RMS} = \sqrt{\frac{1}{T}\int_{t=0}^T p\left( t \right)^2 \mathrm{d}t}\]

where T is the duration of the multi-tone measurement block used in the nonlinear parameter fitting.

The following nonlinear distortion components are calculated:

D_total / Total Distortion (modeled): Total modeled relative multi-tone distortion.
D_bl / Force Factor: Relative distortion contribution of the nonlinear force factor Bl(x).
D_k / Stiffness: Relative distortion contribution of the nonlinear mechanical stiffness K_ms(x).
D_r,int / Internal Mechanical Damping: Relative distortion contribution by the nonlinear internal damping of the suspension materials. This is also called material damping. It includes nonlinear mechanical creep, mechanical hysteresis and nonlinear viscous damping.
D_r,ext / External Resistance Rms(v): Relative distortion contribution of the nonlinear external damping R_ms(v). This depends on the voice coil velocity v and is caused by the air flow resistance near the magnetic gap and in the transducer’s motor. It mainly occurs in small transducers such as micro-speakers.
D_lx / Inductance (x): Relative distortion contribution of the nonlinear displacement-dependent inductance L(x,f).
D_li / Inductance (i): Relative distortion contribution of the nonlinear current-dependent inductance L(i).
D_reluct / Reluctance Force: Relative distortion contribution caused by the reluctance force.
D_sd / Radiation Area Sd(x): Relative distortion contribution of the nonlinear radiation area S_d(x). The identification of this distortion component requires a microphone.
u_DC,model: Asymmetries of nonlinear curves can cause a dynamically generated DC force which shifts the voice coil away from its rest position. This DC force is transformed into the electrical domain resulting in the equivalent DC voltage u_DC,model.

Note

The relative total distortion value D_total is smaller than the arithmetic sum of all distortion components. This is because the distortion components are relative RMS values. If all distortion signals are orthogonal, the total distortion would equal the square root of the sum of the squares of all distortion components.

\[D_\textrm{total,abs} = \sqrt{\sum_i^N D_i^2}\]

However, the distortion components are never orthogonal. They can cancel each other in certain frequency ranges, depending on their phase relationships. Hence, the resulting D_total,abs value is always larger than D_total, which considers the component’s phases.

\[D_\textrm{total} < D_\textrm{total,abs}\]

Errors and Warnings #

No Klippel Server can be reached#

Cause:

The FLSI uploads measurement signals (e.g., voltage and current) to a Klippel server, which analyzes the data to identify loudspeaker parameters and returns the results to your PC. If this server communication fails, an error message is displayed.

Remedies:

Make sure that the Internet works on your PC
- open a website such as https://www.klippel.de in your browser
Make sure that https://klippel.services and its subdomains https://\*.klippel.services are not blocked by your firewall
- open https://nix.klippel.services in your browser
- expected behavior: HTTP error code 4xx (such as 401 or 403)
- if the browser shows an error like Can’t open the page because the server can’t be found or similar, a firewall might be blocking access to the Klippel servers
Make sure that you have the permission to run the software cUrl on the measurement PC
- open a Windows command line (press the Windows button and type cmd)
- type %ProgramData%\Klippel\RnD\Scripts5\Klippel\Library\curl\CURL.EXE "https://nix.klippel.services"
  
  if your dB-Lab installation is not called RnD, substitute RnD with the name of the dB-Lab installation name that you are using
- expected behavior: HTTP error code 4xx (such as 401 or 403)
- if cUrl returns a message like Could not resolve host or an error message with the Windows error code 1260, cUrl might not have the right permissions; in this case, please contact your IT department who needs to permit you to run this software

Make sure that the wiring and clamping is correct#

Cause:

FLSI measured a high internal resistance of amplifier and measurement device. This can be caused by

wrong wiring,
defect or wrong alligator clips, or
loose connections.

Remedies:

Make sure that you have connected all four wires (Sense and Force) of the SPEAKER cable to the transducer terminals.
- Find more information about the speaker cables in the hardware manual.
Alligator clips may fail under continuous use or adverse environmental conditions, such as high humidity.
- Regularly check the alligator clips: The resistance between wire connection point (Speaker cable) and jaws should be \(\ll\) 1 Ohm
- If the alligator clips are worn out and do not hold the wire or transducer connector securely, replace them.
If alligator clips are directly connected to the DUT’s terminals they may become loose during the measurement.
- Consider soldering wires to the terminals and then attaching the alligator clips to the wires.

Spikes or Clipping in the laser signal detected#

Make sure that that the laser is well adjusted and its beam is perpendicular to a bright, reflective measurement spot. It is recommended to create a white reflective coating on the membrane so that the laser has good measurement conditions. If this is not possible, you can deactivate the X target protection limit and specify a safe voltage manually. In this case, the error is displayed as a warning and the measurement is continued even if spikes or clipping is detected in the laser signal.

Check that the correct speaker system type is selected on the property page#

The impedance that was recorded in the small signal measurement during the LINEAR MODE is checked for feasibility. The measured impedance magnitude of a speaker that is expected to have only a single mechanical resonator (e.g., free air or closed box systems) should exhibit only a single resonance peak. This warning is displayed if the measured impedance magnitude exhibits more than one significant resonance peak, which indicates that the wrong speaker system type is selected on the property page. However, additional resonances can exist. For instance, mechanical modes on the DUT’s membrane might cause additional resonances in the electrical impedance.

Note

If you are sure that you have selected the correct speaker system type, please disregard this message.

High Fitting Error (LINEAR MODE / NONLINEAR MODE)#

This warning appears if the linear or nonlinear fitting error exceeded a certain threshold. Make sure that

the correct speaker system type (closed box, vented, …) is selected
the DUT can be described by a second or 4th order mechanical system (maximum of two distinct resonance peaks in the electrical impedance magnitude)
the DUT’s resonance frequency and DC resistance is supported by the FLSI (see the specification document)

Please contact the Klippel support in case if the above statements are fulfilled.

Appendices #

FAQ #

Why do I get different results with the FLSI compared to the LSI3 or LPM?#

The speaker model was improved and comprises new linear and nonlinear parameters, including the nonlinear mechanical creep and the frequency-dependent inductance nonlinearity.

This enhances the accuracy of the existing linear parameters, such as resonance frequencies and Q-factors, as well as nonlinear parameters, such as Bl(x) and nonlinear inductance.

The nonlinear inductance model was improved and extended.

The LSI2 and LSI3 algorithms identify a single nonlinear inductance curve \(L_\mathrm{e}(x)\), assumed to be valid across the full excited bandwidth. As a result, the identified inductance represents an average over the entire excited frequency range. However, this simplification becomes inaccurate when shorting material is present in the transducer motor and/or when the transducer operates over a wide frequency bandwidth. To address this, the FLSI extends the nonlinear inductance model by a frequency-dependent nonlinearity.

In addition, FLSI employs a more insightful representation of the nonlinear self-inductance. Whereas LSI2 and LSI3 display the nonlinearity of the pure inductance \(L_\mathrm{e}(x)\), FLSI calculates the impedance magnitude of the inductance \(|\underline{Z}_\mathrm{L}(f,x)|\), which is directly related to distortion behavior. The advantages of this representation are discussed in greater detail in section Lossy Inductance (x).

The moving mass (see Mechanical and Acoustical Calibration) is identified in the large signal measurement.: This prevents noise in the laser signal from degrading the accuracy of the Bl(x=0) estimate. The result may differ slightly due to varying measurement conditions - for example, a shift in the voice coil position during the large signal test.

What makes FLSI linear parameters more accurate than LPM parameters?#

The FLSI also employs a curve-fitting algorithm similar to LPM. The improved accuracy of the FLSI linear parameters is primarily due to:

FLSI uses a more complex inductance model, improving fitting accuracy of the electrical impedance over the full measurement bandwidth.
FLSI identifies creep and/or leakage effects (in closed-box configurations) directly in the electrical impedance fitting, whereas LPM uses laser data — this improves fitting accuracy of the electrical impedance and parameter accuracy.
FLSI determines stimulus bandwidth, level, and averaging parameters automatically — in contrast, LPM requires manual configuration. Manual setup is more error-prone, as software feedback may not cover all cases of suboptimal settings.
FLSI identifies the moving mass (see Mechanical and Acoustical Calibration) in the large signal measurement, preventing noise in the laser signal from degrading the accuracy of the Bl(x=0) estimate.

Privacy Policy #

What kind of data is stored?#

CONNECTION DATA: IP address, upload time, KA3 number, measurement ID, number of activities

TECHNICAL DATA: measurement signals, measurement setup information, measurement results

CONNECTION and TECHNICAL DATA are stored independently. The TECHNICAL DATA is stored anonymously and cannot be mapped to a specific user.

What is the reason for data storage?#

CONNECTION DATA: Required for server security

TECHNICAL DATA: Statistics of technical data used for bugfixing and algorithm improvements

How long is the data stored?#

CONNECTION DATA: 2 weeks

TECHNICAL DATA: unlimited

Who has access to the data?#

Authorized KLIPPEL employees (IT department and FLSI developers).

FLSI – Fast Large Signal Identification

On this page

1. Specify the Hardware Configuration#

2. Select the Speaker System Type#

3. Select the Protection Parameters#

Parameter Identification#

Measurement Conditions#

Mounting#

Loudspeaker systems#

Measuring Sd(x)#

Measurement Procedure#

Electrical Subsystem#

Mechanical Subsystem#

Frequency-Domain Representation#

Self-Inductance#

Nonlinear electro-mechanical transduction#

Mechanical Resonator#

Mechano-acoustical transduction#

Acoustic Load#

What is mechanical calibration?#

Automatic and Manual Mechanical Calibration#

Added Mass Method#

Acoustical Calibration#

Electrical Impedance#

Lossy Inductance#

Mechanical Stiffness Km(f,x=0)#

Displacement Frequency Response#

Check Relevance of Nonlinear Parameters#

Voice Coil Rest Position#

Force Factor Bl(x)#

Mechanical Stiffness Km(x)#

Lossy Inductance (x)#

Flux Modulation#

Mechanical Resistance Rms(v)#

Symmetry Ranges#

Bl(x) Symmetry Range#

Km(x) Symmetry Range#

Bl(x) and Km(x) Symmetry Points#

Distortion Components#

Displacement Limits#

Thermal Parameters#

Nonlinear Parameters#

Identification Accuracy#

General#

Fitting Error (LINEAR MODE)#

Displacement Fitting Error (LINEAR MODE)#

Fitting Error (NONLINEAR MODE)#

Displacement Fitting Error (NONLINEAR MODE)#

Waveforms and Spectra#

Displacement Waveform and Spectrum#

Distortion Components#

No Klippel Server can be reached#

Make sure that the wiring and clamping is correct#

Spikes or Clipping in the laser signal detected#

Check that the correct speaker system type is selected on the property page#

High Fitting Error (LINEAR MODE / NONLINEAR MODE)#

Why do I get different results with the FLSI compared to the LSI3 or LPM?#

What makes FLSI linear parameters more accurate than LPM parameters?#

What kind of data is stored?#

What is the reason for data storage?#

How long is the data stored?#

Who has access to the data?#

Books#

KLIPPEL Documents#

KLIPPEL Papers#

Papers of other authors#

Standards#