Application Note

Brel & Kjr B K7/6-'89

Accurate Determination of Loudspeaker Parameters usingAudio Analyzer Type 2012 and Laser VelocityTransducer Type 3544by Sren Jnsson, Brel & Kjr, Denmark

1. Introduction

Over the last two decades, it has be-come almost an industrial standardto measure the parameters of a loud-speaker by the method introduced byA.N. Thiele and Dr. Richard Small [1& 2].

Since the method was introduced,more research in loudspeaker model-ling has been made, (e.g., [3, 4 & 5],to mention some), and the measure-ment technique available with newelectronic equipment has advancedtremendously.

The technology of today makes itpossible to find a more accurate andfast method to measure the parame-ters of a loudspeaker.

The method presented here is acompilation of the traditional methodand the method introduced byJ. N. Moreno [6]. It uses an improvedmodel of the loudspeaker and takesvarious non-linearities in frequencyinto account. The method producesaccurate results regardless of thetype of driver used.

The voltage across the driver termi-nals, the current in the voice coil andthe velocity of the diaphragm aremeasured with the driver in free air.The measured results are used to gen-erate the two transfer functions: Thevelocity to current ratio and the imped-

ance. From these functions the loud-speaker parameters can be calculated.

Applying the post-processing capa-bilities of the analyzer to the meas-ured data, it is shown how to correctthe loudspeaker parameters and ex-tract information about the frequencydependent behaviour of some of thecomponents in the equivalent circuitof the loudspeaker.

Since this analysis implied a lot ofarithmetic operations and calcula-tions, an autosequence program forthe analyzer was written. The scopeof the automation process is discussedand measurements of a typical driverare included.

A method to determine the parame-ters characterising low frequency, mid-range and high frequency loudspeakerunits is presented. A laser velocitytransducer is used to detect the veloc-ity of the diaphragm. All measure-ments are made with the loudspeakerin free air. The method uses an im-proved model of the loudspeaker andtakes into account the frequency de-pendent behaviour of some of the ele-ments. It produces accurate resultsand is easy to implement comparedwith conventional methods. The proce-dure is fully automated using an au-tosequence program for the analyzer.The process of automation is discussedand results of a typical driver are in-cluded.

22. Loudspeaker equivalentcircuit

The model of a loudspeaker in freeair used in this note is shown inFig. 1.

The electrical and mechanical sideis separated by a gyrator. A gyratorhas the characteristics that viewedfrom one side, a dual of the networkon the opposite side is seen. It allowsan impedance analogy to be used forthe mechanical side of the circuit.This is a convenient approach, sincethe current Vd flowing through thecomponents MMS, RMS and CMS onthe mechanical side of the circuit isequivalent to the velocity of the dia-phragm. The velocity is measured di-rectly with the laser velocitytransducer.

On the electrical side, the compo-nents RED() and LVC() representthe frequency dependent behaviour ofthe voice coil inductance due to the

eddy currents in the pole structure,as described in [3] and [4]. To obtainaccurate results, the presence ofthese elements are taken into ac-count.

In the model, the radiation resist-ance is neglected while the radiationreactance is included as part of themoving mass (air load).

The radiation resistance is smallenough to be neglected without loss

of accuracy. As an example, for a typ-ical 10 inch woofer in free air, themechanical radiation resistance at100 Hz for both sides of the dia-phragm is more than a 1000 timessmaller than the mechanical resist-ance RMS of the suspension.

The air load for the same woofer isonly about 10 times smaller than themechanical mass MMD of the dia-phragm and is therefore included.

920767/1e

Eg

RED() LVC()RE

+

Es

+

Rg RMSBl

MMSCMSVd

BlVd BlI + +

I

Fig.1 Impedance type model of a loudspeaker in free air

3. Determination of theparameters

3.1 REThe DC resistance of the voice coil ismeasured with a precision bridge.

3.2 fs and QMSTo find the resonance frequency andthe mechanical Q of the driver, themeasured velocity Vd of the dia-phragm is divided by the current I inthe voice coil by postprocessing in theanalyzer. The frequency response ob-tained has the characteristic shapeseen in Fig. 2.

An analysis of the equivalent cir-cuit in Fig. 1 shows that this responsecorresponds to the admittance of themechanical system times a constant,namely the force factor Bl

(1)

Rearranging and letting s = j gives

(2)

VdI

j( ) Bl 1jMMS RMS

1jCMS

+ +=

VdI

s( ) Bl 1MMS

s

s2RMSMMS

s1

MMSCMS+ +

=

Comparing this to a general transferfunction form G(s) expressed in termsof the resonance frequency n andthe quality factor Q

(3)

where k is the gain factor, yields

G s( ) k s

s2n

Qs n

2+ +

=

(4)

and

(5)

where B is the 3dB bandwidtharound n.

From the characteristics of (3) itfollows that:

n1

MMSCMS=

Qn

B=

Fig.2 Magnitude and phase frequency responses for the ratio of diaphragm velocity todriver current. The cursor is placed at the resonant frequency in the phase curve

31. the resonance frequency of thedriver is the frequency at whichthe magnitude of the velocity tocurrent ratio is maximum and thecorresponding phase is zero.

2. the mechanical Q of the driver,QMS can be calculated from (5).

The velocity Vd and the current I areboth present on the mechanical sideof the equivalent circuit in Fig. 1. Asit appears from equation (1), their ra-tio does not include any inductanceor resistance from the voice coil, itonly contains information about themechanical properties of the driver.The resonance frequency and the me-chanical Q found from this ratio aretherefore always accurate regardlessof the type of driver unit.

3.3 QESQES can be calculated as in [2]

(6)

where r0 is the ratio of voice-coil max-imum impedance to DC resistancegiven by

(7)

The definition of QES is the driver Qat fs considering electrical resistanceonly. A small error is therefore intro-duced when calculating r0 from (7),since ZVC contains the reactance dueto the voice-coil inductance. This ef-fect can be removed by using the realpart of ZVC in (7) rather than themagnitude.

Another error is introduced sincethe electrical resistance RED(s ) inFig. 1 is not included in (7). As REDis in series with RE, it will cause QESto rise. The value of QES calculatedusing (6) will therefore be lower thanthe actual QES of the driver.

With both terms taken into ac-count, the following corrected expres-sion for QES is obtained

(8)

3.4 BlThe force factor Bl can be calculatedfrom knowledge of the voice-coil im-pedance and the velocity-to-current

QESQMSr0 1

=

r0ZVC s( )

RE=

QESQMS

ZVC s( ) RealPartRE RED s( )+

1

=

ratio at resonance. The impedance isobtained by dividing the measuredvoltage across the driver terminals bythe current in the voice coil. SeeFig. 3.

At resonance ZVC is, according toFig. 1, given by

(9)

and the velocity to current responseis given by

(10)

RES is the resistance seen from theelectrical side of the circuit in Fig. 1due to the mechanical resistanceRMS.

It can be seen from (9) that theelectrical resistance RES can be iso-lated from the total voice coil imped-ance, by subtracting the DCresistance RE, the resistance due tothe eddy currents RED(s ) and thereactance of the voice coil s LVC(s ).

Subsequently the force factor is ob-tained if RES is divided by the veloc-ity-to-current at resonance:

ZVC js( ) RE RED s( )+=

jsLVC s( ) RES++

VdI

js( ) Bl1

RMS

RESBl

= =

(11)To calculate Bl from (11) and QESfrom equation (8), the resistanceRED(s ) must be known. By initiallyletting RED(s ) equal zero, however,estimates of Bl and QES can be ob-tained and used to calculate the restof the parameters.

Section 4 shows how RED() can bedetermined, allowing a more correctestimation of the other parameters.

3.5 MMS , RMS , and CMSOnce QMS, QES, Bl and RED(s ) areknown, the mechanical parameters ofthe loudspeaker can be calculatedfrom the following equations:

(12)

(13)

(14)

BlZVC js( )[ ] RealPart RE RED s( )+( )

VdI

js( )=

MMSQES Bl( ) 2

s RE RED s( )+( )=

RMSMMSs

QMS=

CMS1

QMSRMSs=

Fig.3 Magnitude and phase of the total driver voice coil impedance. The cursor is placedin the phase curve at the resonance frequency obtained from the phase curve of the velocityto current ratio in Fig.2. The phase difference of about 1 degree compared with Fig.2 iscaused by the inductance of the voice coil

43.6 VAS , SD , and 0Once CMS is known, VAS, the volumeof air having the same acoustic com-pliance as the driver suspension, canbe calculated from

VAS = 0c 2 CMS S 2D (15)

where

(16)

SD is the effective surface area ofthe diaphragm, and d is the diameterof the diaphragm across the loud-speaker cone including half the widthof both sides of the suspension.

It should be mentioned, however,that the accuracy of this method de-pends on the shape and the materialof the suspension. If, for example, thematerial has a non-uniform stiffnessor the width of the suspension con-stitutes a large part of the overallcone diameter, it will affect the accu-racy. In such cases and whenever un-certainty occurs, SD can be obtainedas follows:1. Mount the driver in a closed box

and measure the frequency re-sponse for the ratio of the dia-phragm velocity to the drivercurrent.

2. Use the procedure in [5] to calcu-late the compliance ratio of thesystem.

3. Find VAS as VAS = VB (17)4. Calculate SD as

(18)

Finally the acoustic reference effi-ciency is calculated as in [2]:

(19)

4. Determination of RED()and LVC ()

The impedance of the voice coil, withthe mechanical impedance due to thediaphragm mass, damping and com-pliance removed, is given by

ZVC = RE + RED () +j LVC () (20)

SD d2

4=

SDVAS

0c2CMS

=

042

c3

fs3VASQES

=

or, in accordance with Fig. 1:

(21)

Remember that ZVC is the total elec-trical impedance seen from the ter-minals of the driver.

Since both the impedance ZVC andthe velocity-to-current ratio Vd/I inequation (21) are known from the

ZVCEs B lVd

IZVC Bl

VdI

= =

measurement, ZVC can be calculatedif Bl is known.

Initially, an estimate of Bl is calcu-lated using equation (11), by lettingRED(s ) equal to zero.

Then ZVC is calculated using equa-tion (21) and separated into its realand imaginary parts. The real partthen represents RE + RED() and theimaginary part represents LVC().

RED(s ) is determined by subtract-ing RE from the real part of ZVC ats.

Fig.4 Real and imaginary parts of the voice-coil impedance, with the mechanical im-pedance due to the diaphragm mass, damping and compliance subtracted. The real partrepresents the voice-coil resistance RE+RED(), and the imaginary part represents thevoice-coil reactance LVC ()

Fig.5 Imginary part of ZVC divided by , showing variations in the voice-coil inductanceas a function of frequency

5The whole process is now repeatedby using the value obtained forRED(s ) to correct Bl from equation(11) and so on.

All the arithmetic operations nec-essary to calculate ZVC from equa-tion (21) are made by the analyzer.The result for the driver sample usedin this note is shown in Fig. 4.

If the imaginary part of ZVC i.e.the reactance LVC(), is divided by, the variance of the actual voice-coilinductance with frequency can bestudied. The result is shown in Fig. 5.

The behaviour around 62 Hz inFig. 4 and Fig. 5 occurs at the driversresonance frequency when the oper-ations in equation (21) are per-formed. It may be caused by voice coilheating or time variance, or a combi-nation of these effects during themeasurements. This phenomenon isa subject of further investigations.

5. Further Applications

This section shows in more detailhow the post-processing capabilitiesof the Audio Analyzer Type 2012 areused to manipulate the measureddata.

5.1 Model improvementHaving determined all the loud-speaker parameters of Fig. 1, it wouldbe interesting to know how well theestablished model describes the actu-al behaviour of the loudspeaker.

In section 4, we have already re-vealed the frequency dependent be-haviour of the voice-coil impedanceand presented this in terms of thecomponents RED() and LVC() onthe electrical side of the circuit inFig. 1. It still remains, however, tocheck the mechanical side of the mod-el, where the mechanical impedanceis assumed to consist of the threecomponents MMS, RMS and CMS inseries.

If we return to section 3.2, we havealready measured (the reciprocal of)this impedance with the Laser Veloc-ity Transducer Type 3544 and the Au-dio Analyzer Type 2012. Again bymeans of arithmetic operations wecan simply check the measured resultagainst the model. First the mechan-ical impedance ZM (divided by theconstant Bl) is given by the reciprocalof equation (1):

Fig.6 Real part of the current to velocity ratio, showing the variations in the mechanicalresistance RMS as a function of frequency

Fig.7 Diaphragm displacement of the driver in Table 1 obtained from dividing themeasured diaphragm velocity by the complex frequency s. The peak displacement is shownfor a driver input voltage of 0.5 V

6(22)

Taking the real part and multiply-ing with Bl yields

(23)

When the operations in this equationare performed on the measured data,the expected result should be astraight line with the constant levelof RMS.

The result is shown in Fig. 6. Thevalue of the Bl product is taken fromTable 1. It is not possible to approxi-mate the curve with a straight line.Hence the model in Fig.1 needs to becorrected on the mechanical side tomake it fit the mechanical impedanceof this particular loudspeaker sample.

To find a general proposal as to howthe model should look would requiremeasurements on several differentloudspeaker samples [5].

5.2 Diaphragm displacementBy integrating the diaphragm veloc-ity, the diaphragm displacement canbe obtained. This operation can beperformed easily using the analyzer.See Fig. 7.

6. Measurement

6.1 Measurement set-upA block diagram of the measurementsetup is shown in Fig. 8. The loud-speaker was mounted on the specialbrass clamp as illustrated on thefront page. The clamp ensures thatno vibrations or magnetic leakage oc-cur. It was designed so that theacoustic loading caused by its struc-ture was minimized.

An external amplifier is used topower the loudspeaker.

The voltage across the terminals ofthe loudspeaker, the current in thevoice coil and the diaphragm velocityas a function of frequency is meas-ured in order to perform the calcula-tions described in this note.

The Laser Velocity TransducerType 3544 detects the velocity of thediaphragm. The laser beam is point-ed at the dust cap or as close as pos-

ZM1

Bl

I

Vdj( )=

jMMS RMS1

jCMS+ +

1Bl

=

IVd

j( )RealPart

Bl RMS=

sible to the centre of the cone. Toensure the laser beam is reflected, apiece of retro-reflective tape is fixedat the spot where the beam hits thediaphragm. The angle of incidence ofthe laser beam must be perpendicu-lar to the direction of motion of theloudspeaker cone.

The Audio Analyzer Type 2012 per-forms the measurements and post-processing. A linear sweep with in-

creased resolution around the reso-nance is used for the measurements.

6.2 AutomationSince the analysis given here impliesa lot of arithmetic operations and cal-culations, an automation of the proc-ess is almost a must.

To accommodate this, an applica-tion disk for the Audio Analyzer waswritten.

930283/2e

Audio Analyzer2012

Application DiskWT 8093

Laser Velocity Transducer3544

To RemoteControl

Common

PowerAmplifierWQ 1105

Loudspeaker

Input 1

Input 2

SwitchboxWB 1360

CurrentSense

VoltageSense

DirectBalanced

AdaptorJP 0736

Preamp

Fig.8 Block diagram of the measurement setup

Parameter TraditionalMethodNew

MethodNew method

corrected Units

SD 0.055 0.055 0.055 m2

RE 4.87 4.87 4.87

RED(fs) 0.00 0.25 fs 62.40 62.23 62.23 Hz

QMS 4.81 4.65 4.65

QES 0.49 0.47 0.50

QTS 0.44 0.43 0.45

Bl 16.34 16.90 16.82 T m

MMS 69.7 70.6 70.2 g

RMS 5.55 5.93 5.90 Kg/s

CMS 94 93 93 m/N

VAS 40.1 39.2 40.2 l

0 1.98 1.99 1.89 %

LVC(fs) 2.5 2.5 mHTable1 Loudspeaker parameters of the 12 test driver obtained by the traditional andnew methods with and without corrections for the effects of the eddy currents

7The autosequences on the applica-tion disk guide the user through themeasurements. It performs all thearithmetic operations including acompensation for the phase shift inthe laser velocity transducer. It dis-plays the important curves andpresents the loudspeaker parameterson the screen of the analyzer. A se-lected curve typically the electricalimpedance and the loudspeaker pa-rameters can all be printed or export-ed to a spread sheet for reportgeneration.

Table 1 shows the parameters forthe 12 test driver used in this note.It is a high power low frequency driv-er typically used in vented box sys-tems. The table also includes acolumn showing the parameters ob-tained by the traditional method.

The difference between the tradi-tional and the new method is mostnoticeable for the mechanical Q of thedriver. This effect, of course, also in-fluences the value of RMS.

6.3 Measurement rangeMost typical drivers have a QMS high-er than one, and by inspection of thedriver suspension and diaphragm itis often possible to guess approxi-mately where the resonance frequen-cy will lie.

Assuming that QMS is greater thanone and that the resonance frequencyfs is known, then calculating andcombining equations (3) and (5), theminimum frequency range F and themeasurement start and stop frequen-cy f1 and f2 required to obtain theparameters will be given by

(24)

(25)

(26)

If for example fs is 20 Hz, then Fis 20 Hz, f1 is 12 Hz and f2 is 32 Hz.

Since information above resonanceis also relevant to obtain the imped-ance for example, a much wider fre-quence is typically used in practice.

6.4 Drive conditionsThe current in the voice-coil is meas-ured as a voltage drop across a smallresistance.

A low value of about 1 ohm makesthe loudspeaker operate with almostconstant voltage drive. A higher val-ue of about 500 ohm would make it

F fs=

f1 fs 1.25 0.5( ) 0.6fs=

f2 fs 1.25 0.5+( ) 1.6fs=

operate with almost constant currentdrive.

The value of the resistance can bechanged to any desired value withoutaffecting the accuracy of the meas-urements, since both the voltageacross the loudspeaker terminals andthe current in the voice coil are meas-ured. This provides a simple methodto check the sensitivity of the loud-speaker parameters using differentdrive conditions.

The parameters are, in theory, notexpected to change with the driveconditions. Fig. 9 however shows thatthe velocity-to-current ratio meas-ured using almost constant currentdrive differs from the same one meas-ured using almost constant voltagedrive. The response obtained usingalmost constant voltage has the high-est QMS. The same effect will appearin the impedance curve.

Since in practice, a loudspeaker,usually always operates with almostconstant voltage drive, such condi-tions were used for the measure-ments presented here.

A very low amplifier output mustbe used in order to guarantee linearbehaviour of the driver. A level of0.5 V was used.

7. Conclusion

By measuring the velocity-to-currentratio frequency response, the me-

chanical admittance of the drivers di-aphragm and suspension is obtained.This response is completely free ofany electrical resistance and reac-tance from the voice coil. It providesa very accurate way to calculate QMSand fS, and makes is possible to studythe behaviour of the mechanical pa-rameters of the driver.

Information about the frequencydependent behaviour of the voice-coilinductance and resistance can be ob-tained by subtracting the mechanicalimpedance from the total electricalimpedance of the voice coil. The in-ductance LVC() and an equivalentresistance RED() due to the eddycurrents can thus be determined.Both components were determinedand it was shown how to use the re-sult to correct the expression for QESof the driver.

Using the analyzers arithmetic op-erations with the measured data, themechanical impedance of the loud-speaker was examined. It was foundthat the traditional loudspeaker mod-el needed to be corrected to make itfit the mechanical impedance of theloudspeaker sample used in this note.

It was also shown how the displace-ment of the diaphragm from the ve-locity can be determined.

With the measurement techniqueand postprocessing capabilities of theAudio Analyzer Type 2012 a very ad-vanced tool is available. Much appar-ently hidden information can beextracted from the measured data.

A: Frequency Response, Magn

2 5 10 20 50 100 200Hz0

1

2

3

4

5

m/s/A

920766/1e

Vel/Cur

Fig.9 Magnitude of the velocity-to-current ratio under different drive conditions. The topcurve, with the highest peak at resonance, is measured using almost constant voltagedrive. The other curve is measured using almost constant current drive. The top curvehas a higher QMS than the lower. The driver is a typical 10 woofer suitable for use inclosed box systems

WORLD HEADQUARTERS: DK-2850 Nrum Denmark Telephone: +45 42 80 05 00 Fax: +45 42 8014 05 e-mail: info@bk.dk

Australia (02) 9450-2066 Austria 00 43-1-865 74 00 Belgium 016/44 92 25 Brazil (011) 246-8166 Canada: (514) 695-8225 China 10 68419 625 / 10 6843 7426Czech Republic 02-67 021100 Finland 90-229 3021 France (1) 69 90 69 00 Germany 06151/8149-0 Holland (0)30 6039994 Hong Kong 2548 7486Hungary (1) 215 83 05 Italy (02) 57 60 4141 Japan 03-3779-8671 Republic of Korea (02) 3473-0605 Norway 66 90 4410 Poland (0-22) 40 93 92 Portugal (1) 47114 53Singapore (65) 275-8816 Slovak Republic 07-37 6181 Spain (91) 36810 00 Sweden (08) 71127 30 Switzerland 01/940 09 09 Taiwan (02) 713 9303United Kingdom and Ireland (0181) 954-2366 USA (800) 332-2040Local representatives and service organisations worldwide

Brel & Kjr B K7/6-'89

BO 0384 12 96/08

Since the Laser Velocity Transduc-er Type 3544 measures directly onthe mechanical side of the loudspeak-er, without influence of the voice-coilimpedance, mid-range as well ashigh-frequency units can be meas-ured.

As no known auxiliary complianceor mass are needed, this method iseasier to automate than the tradi-tional method.

Acknowledgement

The author would like to thank JorgeN. Moreno for his great assistanceand dedication to this work. Also,special thanks to A. N. Thiele for tak-ing the time to read the manuscriptand for his comments and sugges-tions.

Finally thanks to Peerless and JBLInternational European Division forproviding the product samples.

Glossary of symbols

Eg Source voltage

ES Voltage across speaker termi-nals

RE Voice coil DC resistance

RED() Voice coil resistance due tothe eddy currents (a functionof )

LVC() Voice coil inductance (a func-tion of )

I Current in the voice coil

Bl Force factor (the product ofmagnetic flux density andcoil length)

Vd Velocity of the diaphragm

MMS Mechanical mass of the dia-phragm including the voice-coil and air-load mass

MMD Mechanical mass of the dia-phragm including the voice-coil but excluding the air-load mass

RMS Mechanical resistance of thedriver suspension

CMS Mechanical compliance of thedriver suspension

VAS Volume of air having sameacoustic compliance as thedriver suspension

VB Net internal volume of enclo-sure

RES Electrical resistance due todriver suspension losses

(Bl)2/RMS

SD Effective surface area of thediaphragm

ZVC Total electrical impedanceseen from the terminals ofthe driver

ZVC Electrical impedance of thevoice-coil, with the mechani-cal impedance due to the di-aphragm mass, damping andcompliance subtracted

0 Reference efficiency

c Velocity of sound in air(344 m/s)

o Density of air (1.18 kg/m3)

References

[1] A. N. Thiele, Loudspeakers in Vented Boxes: Part 1 & 2, J. Audio Eng. Soc., Vol. 19, No 5 (May 1971) & No 6(June 1971)

[2] Richard H. Small, Direct-Radiator Loudspeaker System Analysis, J. Audio Eng. Soc., Vol. 20, No. 5 (June 1972).[3] J. Vanderkooy, A Model of Loudspeaker Driver Impedance Incorporating Eddy Currents in the Pole Structure,

J. Audio Eng. Soc., Vol 37 (March 1989)[4] J. R. Wright, An Empirical Model for Loudspeaker Motor Impedance, J. Audio Eng. Soc., Vol. 38, No. 10 (October

1990)[5] Sren Jnsson, Jorge N. Moreno, Henning Bg, Measurement of Closed Box Loudspeaker System Parameters

using a Laser Velocity Transducer and an Audio Analyzer, Presented at the 92nd Convention of the AudioEngineering Society, March 1992, Vienna

[6] Jorge N. Moreno, Measurement of Loudspeaker Parameters Using a Laser Velocity Transducer and Two-ChannelFFT Analysis. J. Audio Eng. Soc., Vol. 39, No. 4 (April 1991)