Measurement System to Identify Linear and Non-Linear
Characteristics of Loudspeaker Drivers,
Design and Construction
Mario Stoll
June 8, 2013
Abstract
Loudspeaker system developers often use simulation tools to design a loudspeakerdevice. One part is to simulate the low frequency characteristics of the loudspeaker,using a model of the enclosure and the built in loudspeaker driver. Traditionally sucha model of the driver is based on the ”Thiele-Small” parameters, which describesthe electroacoustic behavior in the linear range of a driver. However, in reality, theloudspeaker driver is almost never used in the linear range.
This report describes a system with the goal to measure a single loudspeakerdriver unit to identify the linear parameters and the dominant non-linear parameters.It covers a definition of the loudspeaker parameters, the mechanical set up of themeasurement system with a laser displacement meter and the measurement process.
The built system allows a deeper analysis and faster measurements not only ofone single driver, but complete devices and passive radiators.
1
Contents
1 Introduction 31.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Loudspeaker driver theory and model 52.1 Mechanical construction of a loudspeaker driver . . . . . . . . . . . . . . . 52.2 Sound radiation of a loudspeaker driver . . . . . . . . . . . . . . . . . . . 62.3 Linear Lumped parameter model . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Mechanical part . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Electrical part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Complete model of the driver . . . . . . . . . . . . . . . . . . . . . 92.3.4 Parameters of the linear model . . . . . . . . . . . . . . . . . . . . 11
2.4 Non-linearities in loudspeaker drivers . . . . . . . . . . . . . . . . . . . . . 132.4.1 Non-linear stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Non-linear force factor . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Non-linear self-inductance . . . . . . . . . . . . . . . . . . . . . . . 14
3 Measurement setup 163.1 Overview of the hardware setup . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Voltage and current . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Laser displacement sensor . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 Output signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.4 Recommendation for measurement power . . . . . . . . . . . . . . 173.1.5 Driver placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.6 Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Measurement process 194.1 Input driver parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Data storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Measurement results and parameters 235.1 Validation of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Example Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 Conclusion 286.1 Current status and further steps . . . . . . . . . . . . . . . . . . . . . . . 28
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1 Introduction
Sonic Emotion offers sound entertainment solutions in different fields like audio spatial-ization, sound enhancing and loudspeaker optimization. One part is the acoustic designof loudspeaker enclosures paired with customized audio signal processing. For this work,every component in the audio path is tuned, where as the loudspeaker driver has asignificant influence to the whole system.
To get the best result, it’s crucial to know as much as possible about the loudspeakeritself, and how exactly to use it in the environment. This report covers the design of ameasurement system used to identify the behavior of a loudspeaker driver in the linearand non-linear range. Those measurements are later used to design the loudspeakerenclosure (good low frequency response) and the signal processing part (different soundsettings depending of the playback level). The final product - is it ”high end” musicproduct or a sound system in a TV - the driver works within the defined boundariesand provides an optimal sound output. Further, based on the parametric model, analgorithm can be applied to limit the non-linear behavior.
Previously, a set of different tools has been developed, including a simulation softwarethat allows to simulate linear and non-linear behavior for multiple drivers in a sharedenclosure [1]. So far this model was fed with linear parameters of a loudspeaker driver,which are usually provided by the loudspeaker supplier or they can be measured with awell known impedance measurement. The new measurement system will allow a betteranalysis of the linear and non-linear behavior, but also it will be able to perform nonexpensive and fast measurements on drivers and complete devices. Further, it will alsobe capable to measure micro speakers and passive membranes, and - by measuring theenclosure - vibration issues of the loudspeaker box.
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1.1 Previous work
To capture the additional displacement signal input in a good quality, a number of testsand simulations have been performed [2]. The overall outcome of the tests is:
• How to capture the displacement of the cone: In theory, the radiated sound pres-sure level could be captured and used as an input to the system. However, thistechnique didn’t yield good estimations. An other option is to capture the move-ment of the cone with a laser, where as there are different laser models available,which can detect the displacement, the velocity or the acceleration. Due the largeprice difference, a displacement laser sensor offers the best value per cost.
• What kind of stimulus signal: An exponential sweep input signal with waveletde-noising showed best performance in simulations.
• Requirement for the signal to noise ratio: All input signals should have more than50 dB signal to noise ratio, what can be reached with a displacement sensor.
Based on this outcome it was chosen to realize the new measurement system with alaser displacement meter.
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2 Loudspeaker driver theory and model
2.1 Mechanical construction of a loudspeaker driver
Below a typical electrodynamic loudspeaker driver (2 inch broadband) is shown (Fig-ure 1), with its main parts, the voice coil inside permanent magnet, the cone (or mem-brane) and the basket. Figure 2 shows the voice coil from the same driver but withshifted out magnet. Feeding the voice coil with alternating current, the coil moves inthe magnet field due the Lorentz force1. The connected cone transmits this mechanicalmovement into movement of the air around the driver. The basket provides a stablestructure to hold all parts together.
Figure 1: Loudspeaker Driver Figure 2: Open Driver
Figure 3: Moving part and basket
The moving part, consisting of the voice coil and the cone, are hold on place bytwo suspensions, the spider (Figure 3) and the surround. This rigid combination of thespider and the surround allows the cone to move on one axis and act as a circular piston,
1the Lorentz force is the force on a point charge due to electromagnetic fields
5
but also introduces a restoring force to the rest position.For broadband drivers, the material of the cone is usually paper or aluminum to makethe moving mass as low as possible. The example driver above has a aluminum cone.The showed driver is referenced as the 2 inch example driver for this report.
2.2 Sound radiation of a loudspeaker driver
A loudspeaker box is a system which converts an electric signal to a mechanical to aacoustical signal. Two main parts of the system can be defined:
• Part one is the loudspeaker driver with voltage input from the amplifier, where asthe output is the mechanical movement of the cone.
• The second part is the acoustic radiation, with the moving cone as an input.
This movement of the cone can be modeled as a pistonic movement in the so calledpiston range. The piston range is the frequency range where the wavelength is largerthan the cone circumference [3]. This condition can also be written as k · r = 1, withthe radius of the circular cone r and the wavenumber k (k = 2π
wavelength).The product of the velocity v and the area of the cone Sd gives the volume velocity
Q. The radiated sound pressure or the moving cone is depending on Q and the radiationimpedance Zr. Zr is the influence of the surrounding media (usually air) to the coneand a complex quantity with an active and reactive component, where as the active(real part) describes the sound radiation and the reactive (imaginary) part describesthe movement of the air with no sound radiation. For a moving piston in an infiniteextended baffle, the real part of Zr is showing a high pass function first order, the cornerfrequencyfkr=1 is approximately where the wavelength is the cone circumference. Whenthe driver is radiating in free air - means not mounted in a baffle - the high pass showsa function of the 2nd order. For the 2 inch example driver, fkr=1 is at around 2.2 kHz.
Using equivalent electric circuits, the high pass shape of Zr can be modeled with afrequency dependent friction resistance and an inductance. Since the friction loss due theair is negligible compared to the friction loss in the surround and spider, the radiationimpedance can be further simplified using the acoustic mass Mac. The acoustic mass isdefined as the air load seen by the cone which get’s a displacement force.
Below the frequency point fkr=1, the radiation is omnidirectional, while above, apistonic movement of the cone will result in a higher directivity. This means, the soundradiation is more focused to the perpendicular of the cone. Real drivers start to ”brakeup” for higher frequencies, modes on the cone are developed and only the inner partshows a pistonic movement.
2.3 Linear Lumped parameter model
The part one from the classification above - the loudspeaker driver - is forming a dampedmass-spring-resonator coupled to the input voltage through the voice coil. For smallsignals (where the driver operates in the linear range) and operating within the pistonrange, it can be modeled with the linear Lumped parameters.
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2.3.1 Mechanical part
The damping due the friction loss, the restoring force of the suspension and the movingmechanical part of the driver acts as a resonator. The total driving force Fa of thedamped mass-spring-resonator is mainly composed of three parts:
Fa = FR + FM + FK (1)
• Friction Force FR : Friction loss in the mechanical and acoustical elements.
FR = Rm · v (2)
• Inertia Force FM : Mass of the cone, the voice coil, the voice coil former, movingparts of surround and spider, also the air load on the front and backside of thecone Mac.
FM = Mm · a (3)
• Restoring Force FK : Forces the moving parts to the rest position, based on thebehavior of the surround and the spider.
FK = Km · xd (4)
with:xd: displacement of the conev: velocity of the conea: acceleration of the coneThe derivation to time of xd is v, the derivation to time of v is a.
The schema below shows the mechanical set up of a driver forming the damped mass-spring-resonator with the discussed forces (Figure 4).
Figure 4: Mechanical part
7
Fa
FM
Mm
FK
1Km
FR
1Rm
v
Figure 5: Equivalent of the mechanical part
The mechanical part can be modeled to a corresponding electric equivalent circuitusing the ”force to current” analogy (Figure 5)[4].
Out of the circuit, the impedance Zmech can be defined as:
Zmech =Fav
(5)
For Zmech a Resonance Frequency fs can be calculated:
fs =1
2π·√Km
Mm(6)
Three main ranges for different excitation frequencies f can be found [5]:
f << fs Frequency below the resonance frequencyThe stiffness Km of the surround and spider is relevant
f = fs Frequency at the resonance frequencyThe mechanical resistance (due the friction losses) R is relevant
f >> fs Frequency above the resonance frequencyThe moving mass Mm is relevant
Table 1: Frequency ranges of the mechanical system
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2.3.2 Electrical part
The system above is expanded with the motor, i.e. the permanent magnet and the voicecoil. The magnet of the driver generates the magnetic flux density (induction). Whenthe voice coil is fed with a current i, due the Lorentz force, the coil gets a displacementforce Fa. Fa is dependent of the flux density B and the voice coil length l (length of thecoil wire in the magnetic field).
Fa = Bli (7)
Due the velocity v of the voice coil in the magnetic field, a back induced voltage uemf(electro motive force) is generated.
uemf = Blv (8)
The voice coil has a resistance Re and the self inductance Le. The self inductancecan be neglected for low frequencies, but reduces the current at higher frequency.
2.3.3 Complete model of the driver
The mechanical part and the electrical part are coupled with the transformer of thevoice coil, with the coupling constant Bl. Forming a system including a voltage sourceconnected to the driver terminals results in the following circuit:
idRe Le
ud uemf
Fa
FM
Mm
FK
1Km
FR
1Rm
v
Figure 6: Complete electrical equivalent circuit of a loudspeaker driver
For this circuit the input impedance is defined as:
Zdriver =udid
(9)
The picture Figure 7 shows the free air impedance magnitude response of the 2 inchexample driver with its typical shape. There is the influence of the damped mass-spring-system around the resonance frequency fs, where the magnitude is at the maximum.
The displacement xd of the 2 inch sample driver is shown in Figure 8. The stimulussignal level is about 2.5 V RMS and the driver is mounted in free air. The displacementis large for the part below the resonance frequency, where as above it gets small.
Figure 9 shows the sound pressure level measurement in a distance of 0.5 m. On thisreal example, the ranges can easily be detected with fs = 176 Hz and fkr=1 = 2200 Hz.
9
30 50 100 200 500 1000 20000
5
10
15
20
25
30
35
40
Frequency [Hz]
Impedance [O
hm
]
Figure 7: Electrical impedance of the 2 inch example driver, fs is at 176 Hz
102
103
0
0.5
1
1.5Displacement
Frequency [Hz]
Dis
pla
ce
me
nt
[mm
]
Figure 8: Displacement of the 2 inch example driver, fed with a stimulus signal level ofabout 2.5 V RMS
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Figure 9: Sound pressure level measurement of the 2 inch example driver, red: mountedin the ∞-extended baffle, blue: free air
.
Using the equivalent circuit and the radiation pattern of the driver, three frequencyranges can be described:
• Below the resonance frequency fs: The volume velocity Q rises proportional to fand the radiation impedance Zr gets as well a rising proportional to f (for the case∞-extended baffle). The radiated sound pressure level raises with f2.
• Between fs and fkr=1: The volume velocity Q gets a decrease of 1/f and whereas the radiation impedance Zr rises as well to f (for the case ∞-extended baffle).The radiated sound pressure level is constant in this range (independent of f).
• Above fkr=1: Due inductance of the voice coil the impedance gets a small rising,the modes on the cone tends to appear and the cone is no longer moving as a piston.To model this upper frequency range, there are better models than the Lumpedparameter model. Most often, a loudspeaker system designer uses a sound pressurelevel measurement of the real driver to evaluate this frequency range.
2.3.4 Parameters of the linear model
The physical basic parameters describe the low frequency part below fkr=1 according theelectric equivalent circuit and are summarized below, followed by the widely acceptedThiele-Small Parameters (TSP). The TSP are the standard parameters in the industryto characterize a loudspeaker driver. One reason is, because to determine the parametersof the equivalent circuit above, the electrical impedance can be measured at the terminalsof the driver using a voltage source. With the measured impedance plot, some of theparameter values can be read out. For a complete parameter identification, a secondimpedance measurement is performed, but with a slightly modified mechanical setup.There are two common approaches, the ”closed box” (the driver mounted in a closedbox) and the ”added mass” (mass added to the cone) techniques as the modified set
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up. With the combination of the two measured impedances all TSP are defined. Sincefor this measurement the electrical impedance is captured, it’s an easy setup since nofurther equipment like a displacement laser meter is needed.
A full definition of the parameters and a guideline to identify the parameters is in[6] and [7].
The presented setup using the displacement laser sensor, there is no need the performtwo different measurements. With the additional information of the cone behavior, allTSP are determinate.
Electrical Parameters
Re Resistance of the voice coil
Le Voice coil inductance (or self-inductance) of the driver
Be = Bl Force factor
B Magnetic flux density
l Length of the conductor in the magnetic field
Mechanical Parameters
Mm Mass of driver membrane including air load and voice coil
Rm Resistance of driver suspension losses
Km Stiffness of driver suspension
Cm = 1/Km Compliance of driver suspension
Sd Effective projected surface area of driver membrane
xmax Maximum linear displacement
Thiele/Small Parameters
fs Resonance frequency of the driver
Qes Quality factor at fs considering electrical resistances only
Qms Quality factor at fs considering mechanical resistances only
Vas Compliance of the driver expressed as an equivalent air volume
Table 2: Overview of the parameters of the linear model
Analyzing the 2 inch example driver, the parameters show the values Table 3.:
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Re 3.6 Ohm
Le 0.03 mH
Be 2.83 Tm
Mm 1.64 g
Cm 502.3 um/N
Sd 15.2 cm2
xmax 1.65 mm
fs 176 Hz
Qes 0.81
Qms 5.6
Vas 0.16 l
Table 3: TSP of the 2 inch example driver
2.4 Non-linearities in loudspeaker drivers
The Linear Lumped Model - valid only for small cone excursions - represent the realloudspeaker driver as a good basis to perform simulation and optimization for the loud-speaker enclosure design. However, a loudspeaker driver in general is a highly non-lineardevice. Where as the traditional multiway loudspeaker box might not show high non-linear behavior, small broadband drivers used as a example in the modern TV’s mayexceed the linear range quite easy during the normal use. It’s evident, that for the casethe cone gets a large displacement, the parameters show a dependency of the displace-ment. Due to this fact, the linear parameter model is extended with the most significantnon-linearities [5]. First, the source of the significant non-linearities are localized, beforethey are modeled as a Voltera series expansion, with xd as the displacement:
Paramter(xd(t)) = a0 + a1xd(t) + a2xd(t)2 (10)
Using this method, a parabola curve can be described, depending on the displace-ment.
2.4.1 Non-linear stiffness
The surround and the spider show a different behavior for large cone extensions. Spe-cially the surround is stretched for large displacements. Due the material deformationthe stiffness rises and the restoring force is increased. It can’t be modeled as a linearspring anymore, the stiffness is (see also Figure 10):
Km(xd(t)) = km + km1xd(t) + km2xd(t)2 (11)
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−3 −2 −1 0 1 2 3
x 10−3
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3x 10
−3 Compliance Cm
[m/N
]
[m]
Figure 10: Stiffness of a 4 inch driver, Cm depending of excursion, red: linear parametervalue, blue: non-linear parameter curve
.
2.4.2 Non-linear force factor
For the linear model, the magnetic field was expected to be constant over the displace-ment x. But for larger displacements, the electromagnetic driving force is lower, sincethe voice coil don’t get the full flux desitiy anymore, but also the back induced voltage isdifferent (see also Figure 12). The force factor Be is therefore written as (see Figure 11):
Be(xd(t)) = be + be1xd(t) + be2xd(t)2 (12)
−3 −2 −1 0 1 2 3
x 10−3
4.75
4.8
4.85
4.9
4.95
5
5.05
5.1
5.15
5.2
5.25
Force Factor Be
[Tm
]
[m]
Figure 11: Force factor of a 4 inch driver, Cm depending of excursion, red: linearparameter value, blue: non-linear parameter curve
.
2.4.3 Non-linear self-inductance
The non-symmetric construction of the magnet - voice coil system introduces two non-linearity if the inductance to the model.
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The inductance of a coil depends of its shape and its surroundings. Regarding thevoice coil system, the surroundings for the coils is different if the coil is moved to theinner position (surrounded by iron) or to the outer position (surrounded by air). Whenthe voice coil is at the inner position (Position -a), the self-inductance is much higher,than when the voice coil is moved to the outer position (Position +a), see Figure 12.The second non-linearity of the inductance is due the saturation of the iron, which isdependent of the voice coil current [5]. The non-linear self induction is more significantfor large than for small drivers.
Figure 12: Schematic view of the permanent magnet (black), the iron (gray) and thevoice coil for different positions of the displacement xd.
The non-linear self induction is therefore extended as:
Le(xd(t)) = le + le1xd(t) + le2xd(t)2 (13)
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3 Measurement setup
The measurement setup is a computer based system with a connected audio interface toaddress the inputs and outputs.
3.1 Overview of the hardware setup
On the audio interface, one output and three inputs are used:
1. The output is connected through the power amplifier to the driver terminal
2. Input uv is capturing the voltage over the driver terminal
3. Input uv is capturing the current to the driver
4. Input ulaser is connected to the analogue output of the displacement laser sensor
Figure 13 shows the complete electrical setup and its parts are describen in thefollowing chapters.
Figure 13: Complete electrical measurement setup
3.1.1 Voltage and current
To identify the parameters of the driver, the impedance at the driver terminals is mea-sured. The driver is fed with a stimulus signal and the voltage uv and the current arecaptured. Since it’s not possible to measure the current with an audio interface, a smallcircuit is added to transform the voltage uc over a resistor - in serial configuration to thedriver - to the requested current value. The amplifier to power the driver under test isa common audio amplifier, which is designed for low impedance loads, like loudspeakerdrivers usually are. Due this reason, the serial resistor needs to show a low value, whichis 1 Ohm. The voltage over this resistor is small, so the significant influence of noise inthe signal path is addressed with proper and careful cabling.
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3.1.2 Laser displacement sensor
The laser displacement sensor emits a red beam of the size 0.1 * 0.1 mm with a maximumoptical output power of 1 mW. The propagation line is perpendicularly to the case andhits the measurement target in front. The laser beam is then diffused, where as oneportion is reflected back to the laser head. The displacement can then be determined bydetecting the angle of the reflected beam on a position sensitive detector within the laserhead. The measurement distance of the used laser is 30 mm, with a range of ±4 mm anda resolution of 0.5 µm. With a sampling cycle of 200 µs, we can cover the measurementfrequency range. The laser head provides an analogue voltage output proportional tothe measurement distance. Positioning the laser head in 30 mm in front of the cone, thedisplacement is measured. The analogue voltage output is set to fit to the input of theaudio interface. This scaling value can be stored into the laser head. The laser head isa product from Panasonic, the type is: HL-G103-S-J.
Figure 14: Laserhead pointing to the center of the 2 inch example driver.
3.1.3 Output signal
The stimulus signal to perform the measurement is a positive chirp signal with a lengthof 6 s, with the frequency range 20 Hz to 2 kHz. Since the driver needs to be operatedin the linear range, the output level needs to be small. During tests with different drivermodels and output levels, an optimal output voltage level of 250 mV RMS was found toyield good results for the linear parameters.
3.1.4 Recommendation for measurement power
For the large signal measurement the level is set to 2.5 V RMS and 5 V RMS. However,the stimulus signal strength must not drive the driver over the limits and destroy it.Therefore the best case is to adjust the level per driver.
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3.1.5 Driver placement
The model to parametrize is based on the free air radiation of the cone, therefore thedriver has to be installed in the same situation, means the radiation in front and backof the cone must not be influenced. This can be reached by placing the driver into thestand in a vertical position.
3.1.6 Stand
The mounting of the driver needs to meet the requirements of the free air radiation,but also it needs to provide a stiff construction and stable holding for the laser head.Further, the stand should be flexible to possibly mount small devices to measure - as anexample - passive membranes. The driver is first mounted in the vertical position, thenthe laser head is placed in front of the cone. To be fully flexible, at the moment clampsare used to mount the driver and the laser head. In a second step, the mountings canbe developed for a ”one-hand” operation.
Figure 15: Complete laser stand.
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4 Measurement process
In advance of any loudspeaker driver measurement, the driver needs to be ”burnt in”.This is a short amount of time (about 5 minutes), where the moving parts needs todisplaced. This can be done with playing a sine wave with a large amplitude at around75 % of the resonance frequency.
The measurement tool is working on a computer, using the measurement setup de-scribed above (Figure 13) and is running in a Matlab environment. The used library”Playrec” and ”Portaudio” allows to access the audio interface from Matlab.
4.1 Input driver parameters
As a first step, the driver parameters like ”brand” and ”type” need to be filled in, followedby the ”effective cone diameter”. This value is defined as the diameter of the cone itselfplus on both sides half of the surround (Figure 16). The value should correspond to thesize of the moving piston and is used to calculate the acoustic radiation of the driver.
Figure 16: Effective cone diameter, 44 mm for the 2 inch example driver
The last input parameter is the desired measurement level value (0.25, 2.5, 5 VRMS).
4.2 Calibration
When the hardware is set up and the system is started, the calibration of the inputs andoutputs of the audio interface has to be done to make meaningful measurements. Thisprocess takes 3 steps, where the operator is guided through:
1. Calibration of the output signal. A 50 Hz sine wave is played out through theamplifier to a 10 Ohm resistor, which is connected instead of the driver. The voltageover this resistor is measured and should show the defined measurement value. Ifnot, the operator needs to adjust the signal strength. Once the output level is
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good, it will be kept at the same level during the calibration and measurementprocess. 50 Hz is chosen to make the use of a standard measurement equipment(volt meter) possible.
2. Calibration of the input probes. Since the input gains can be set variously, theyare calibrated to allow absolute measurements. A 1 kHz sine wave is played outwith the connected 10 Ohm resistance and the inputs uv and uc are captured. Therecorded levels of uv and uc represent the real Voltages over the 10 Ohm and the1 Ohm resistor. Those factors are stored as the calibration gains.
3. Calibration of the laser head. The reason is that the laser head can be resetto different displacement, but also the input ulaser of the audio interface can bedifferent. For this process a small mechanical setup with a reference position anda reference displacement has been built. Since the connected audio interface isnot able to capture DC-voltages, the reference displacement unit needs to show aAC-voltage. Therefore a motor with a connected elliptic formed disc (Figure 17) isused. Turning on the motor, the laser head captures the displacement of the disc.Since the displacement of the disc is known, the voltage recorded by the audiointerface is referring to the displacement by a correction value.
Figure 17: Complete laser stand
The calibration is done and the calibration values are stored during the measurementsession, where the operator is able to change the output level.
4.3 Measurement
The measurement starts by properly mounting the driver on the stand and placing thelaser head 30 mm in front of the driver. The beam of the laser has to point to themiddle of the cone to allow good reflections of the laser. The used model is based onthe moving piston, therefore it’s assumed that the cone shows the same behavior in the
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measured frequency range. Once the set up is complete, the operator performs one ormore measurements. At the end, each of the three captured time signals is stored forthe following parameter estimation process. As an example, the recorded time signalswith the stimulus chirp of 2.5 V RMS are shown in Figure 19.
0 1 2 3 4 5 6 7 8
−3
−2
−1
0
1
2
3
Timesignal Voltage
Time [s]
Am
plit
ue [V
]
0 1 2 3 4 5 6 7 8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Timesignal Current
Time [s]A
mp
litu
e [
A]
0 1 2 3 4 5 6 7 8
−1.5
−1
−0.5
0
0.5
1
1.5
Timesignal Displacement
Time [s]
Dis
pla
ce
me
nt
[mm
]
Figure 18: 2 inch example driver, stimulus signal with 2.5 V RMS, voltage, current anddisplacement time responses
4.4 Data storage
The interface to the parameter estimation tool is the saved measurement structure. Thestructure contains (Table 4):
Name Brand, Model
Diameter of the cone in meters
Calibration data calibration data of the audio interface,output voltage, correction gains for uv ,uc ,ulaser,
the recorded time signals of the calibration
Measured data time signals for uv ,uc ,ulaser
Table 4: Structure to store the measurements
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4.5 Parameter estimation
Using the electrical equivalent circuit (Figure 6), to following equations can be developed:
ud(t) = Reid(t) + Ledid(t)
dt+Be(t)
dxddt
, (14)
Beid(t) = Mmd2xd(t)
dt2+Rm
dxd(t)
dt+Kmxd(t). (15)
ud(t) is the input voltage at the driver port and id(t) is the input current, which arecaptured with the measured system. xd(t) is the displacement of the driver membrane,which is captured with the laser sensor.
The parameter estimation algorithm using the measured time signals is based onthose equations and on a Discrete Time State Space Model presented in [1].
The process of the parameter evaluation part is:
1. Load the measurement structure
2. Cut the time signals (the measurement process adds a time buffer at the beginningand at the end of the recording).
3. De-noise the displacement signal using wavelet de-noising filter available in Matlab[2].
4. Estimation of the driver parameters using the provided algorithm.
5. Finally the driver model is saved and can be used for visualizations and simulations.
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5 Measurement results and parameters
5.1 Validation of the system
To validate the measurements, the proper functionality of the following parts were re-viewed.
The validation of the measurement system can be divided in to the verification of itssubsystems.
• Electrical signals: On the electrical side, the output level at the driver terminals isverified to fit the given output power. With a scope, the whole measurement fre-quency range was analyzed to provide a proper output through the audio interfaceand the amplifier. A similar procedure with a scope was done to measure the elec-trical inputs uv and uc. Out of this recorded signal, the impedance curve aroundthe resonance frequency fs was calculated and cross checked with the specificationof the driver supplier.
• Laser signals: The laser head provides the used analogue output, but also a digitalinterface. Through this interface, the displacement data can be read out. Withthe calibration disk with the given excursion, the recorded analogue signals isverified with it’s mechanical dimensions and the digital data read from the laserhead. Different disks and example drivers have been checked, all tests showed goodresults. Therefore, also this part of the system is verified.
• Play-out and recording of the signals: The play-out out of Matlab thought ”Playrec”has been tested, as well as the capturing of the incoming signals using this chainshowed good results. The recorded time signals of the different disks and theexample drivers showed its expected behavior.
5.2 Example Measurements
Example measurements are presented below. The first measurements show the resultsof the 2 inch example driver with different stimulus strengths.
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0 1 2 3 4 5 6 7 8
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Timesignal Voltage
Time [s]
Am
plit
ue
[V
]
0 1 2 3 4 5 6 7 8
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
Timesignal Current
Time [s]
Am
plit
ue
[A
]
0 1 2 3 4 5 6 7 8−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Timesignal Displacement
Time [s]
Dis
pla
ce
me
nt
[mm
]
Figure 19: 2 inch example driver, 250 mV RMS. The voltage and current curve showthe expected shape defined by the electrical impedance the amplifier. The displacementof the cone is low, but noisy. Out of this measurements, the linear parameter can beevaluated, but the estimation of the non-linear parameters- specially the stiffness - givesnot good results.
24
0 1 2 3 4 5 6 7 8
−3
−2
−1
0
1
2
3
Timesignal Voltage
Time [s]
Am
plit
ue [V
]
0 1 2 3 4 5 6 7 8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Timesignal Current
Time [s]
Am
plit
ue
[A
]
0 1 2 3 4 5 6 7 8
−1.5
−1
−0.5
0
0.5
1
1.5
Timesignal Displacement
Time [s]
Dis
pla
ce
me
nt
[mm
]
Figure 20: 2 inch example driver, with 2.5 V RMS, the voltage and the current areincreased by the factor 10, also the displacement shows approximately the same factor.However, also some effects due the non-linear behavior is visible. On the displacementfigure, also a small non-symmetry is visible, which may be caused by the inductancenon-linearity.
25
0 1 2 3 4 5 6 7 8
−6
−4
−2
0
2
4
6
Timesignal Voltage
Time [s]
Am
plit
ue [V
]
0 1 2 3 4 5 6 7 8
−1.5
−1
−0.5
0
0.5
1
1.5
Timesignal Current
Time [s]
Am
plit
ue
[A
]
0 1 2 3 4 5 6 7 8
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Timesignal Displacement
Time [s]
Dis
pla
ce
me
nt
[mm
]
Figure 21: 2 inch example driver, he stimulus signal is increased to 5 V RMS, alsothe current is doubled. The displacement can not follow this increase and is limitedto around 2.2 mm. Due the stimulus signal strength, the driver is operating above thelinear cone excursion, which is specified to 1.65 mm.
26
The second example is a 4 inch woofer with a fs at around 45 Hz, but also the ratedmaximal linear excursion is higher. Therefore, the movement of the cone is extendingthe measurement range of the laser sensor. Figure 22 shows three different measuredistances. First - as the standard - placed at 30 mm distance to the cone. The timeresponse shows a limiting of the excursion at low frequency, which is due the limitingof the measurement range of ±4 mm. Changing the measurement distance to 28 mm(plot in the middle) or to 32 mm (right), discovers that the driver’s excursion is higherthan the measurement range. The operator needs to take care of this and the stimulusstrength needs be be decreased.
0 1 2 3 4 5 6 7 8
−5
−4
−3
−2
−1
0
1
2
3
4
5
Timesignal Displacement
Time [s]
Dis
pla
cem
ent [m
m]
0 1 2 3 4 5 6 7 8−6
−4
−2
0
2
4
6
Timesignal Displacement
Time [s]
Dis
pla
cem
ent [m
m]
0 1 2 3 4 5 6 7 8−5
−4
−3
−2
−1
0
1
2
3
4
5
Timesignal Displacement
Time [s]
Dis
pla
cem
ent [m
m]
Figure 22: 4 inch example driver, 5 V RMS, with different distance of the laser head tothe cone.
27
6 Conclusion
6.1 Current status and further steps
A measurement system to determine linear and non-linear parameters has been designedand built, including the required hardware and software. While the hardware consistsmainly of a stand, a laser displacement sensor and an impedance circuit connected to anaudio interface, the software covers the generation of the stimulus signals, the recordingof the measurement values, the data storage and its process flow. The conclusion of thework so far and further steps:
1. The selected laser sensor proved as a good choose, since the specification fits tothis use case and output signals show low noise and good linearity. In practicalwork, the built in display is helpful to position the laser head.
2. The built stand is stiff and provides a stable position of the driver and the lasersensor. However, the positioning of the laser head and the driver can be increased,since the current status with the clamps is not handy. Further mechanical designwork is needed to provide a practical solution for this. Also, for large movingmasses, the driver suppling balk starts to vibrate. This issue will be solved byadding a second balk as a stiffener.
3. The defined output voltage levels of 0.25, 2.5 and 5 V RMS showed meaningfulresults for the few tested example drivers. The practical usage of the system mightprovide an update to those values and a guideline for different measurement levelsper driver sample.
4. On the software side, a parameter estimation process is requested to display theresults in straight after the measurement. In this way, the operator get directfeedback of the driver measurement and might redo it when he discovers any con-tradictions.
5. As a further step, the system will be adapted to measure passive radiators andcomplete devices with multiple drivers.
6. As presented above, depending on a driver and the stimulus strength, the range ofthe driver displacement can extend the measurement range of the laser. Thereforethe measurement distance needs to be 30 mm, the stimulus signal level maybedecreased and the measurements are checked immediately. As a next feature, itmight be evaluated to combine two measurements with a shift in the measurementdistance to cover the full displacement range of a driver.
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References
[1] Matthias Fluckiger. Interaction in loudspeaker arrays interaction in loudspeakerarrays with shared enclosures. Master Thesis at ETH Zurich, 2011.
[2] Nathan Willson. Determining the boundaries of a loudspeaker parameter estimationalgorithm. University of Victoria, 2012.
[3] R. Small. Direct-radiator loudspeaker system analysis. IEEE Transactions on Audioand Electroacoustics, 19(4):269–281, 1971.
[4] Kurt Heutschi. Mechanical and acoustical analogies. Course material of ETH Zurich,2009.
[5] W. Klippel. Loudspeaker nonlinearities-causes, parameters, symptoms. J. AudioEng. Soc, 54(10):907–939, 2006.
[6] Kurt Heutschi. Loudspeakers. Course material of ETH Zurich, 2009.
[7] Wolgang Klippel. Linear lumped parameter measurement. Hands-On Training 1,2013.
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