JAES_V59_1_2_PG44hirez

ENGINEERING REPORTS

Dynamic Measurement of Transducer EffectiveRadiation Area*

WOLFGANG KLIPPEL** AND JOACHIM SCHLECHTER([email protected]) ([email protected])

Klippel GmbH, DE-01309 Dresden, Germany

The effective radiation area SD is one of the most important loudspeaker parameters because

it determines the acoustic output (sound power) and efficiency of the transducer. This

parameter is usually derived from the geometrical size of the radiator considering the diameter

of one-half the surround area. This conventional technique fails for microspeakers and

headphone transducers where the surround geometry is more complicated and the excursion

does not vary linearly with the radius. New methods for measuring SD more precisely are

discussed. The first method uses a laser sensor and a microphone to measure the voice-coil

displacement and the sound pressure generated by the transducer while mounted in a sealed

enclosure. The second method uses only the mechanical vibration and the geometry of the

radiator as measured using a laser triangulation scanner. The reliability and reproducibility of

the conventional and the new methods are verified and the propagation of the measurement

error on the T/S parameters using the test-box perturbation technique and other derived

parameters (sensitivity) is discussed.

0 INTRODUCTION

Electroacoustic transducers can be modeled at low

frequencies by an equivalent circuit comprising a few

lumped parameters. The effective radiation area SD

describes the coupling between mechanical and acous-

tical domains by replacing cone, diaphragm, surround,

and other radiating parts with a rigid piston moving with

the same velocity as the voice coil and having an

equivalent surface generating the same sound pressure in

the far field as the real radiator [1]. The parameter SD

determines the sound pressure on axis (passband

sensitivity) and the total acoustical output power [2]. A

precise value of SD is also a prerequisite for accurate

parameter measurements using the test-box perturbation

technique and for providing reliable T/S parameters

(such as the equivalent air volume of driver stiffness Vas)

for system design [3], [4].

In practice the effective radiation area is usually

derived from the geometry of the radiating surface by

Fig. 1. Estimation of effective radiation area SD from

radiator geometry.

*Manuscript received 2010 June 2; revised 2010 October 28.**Also with University of Technology Dresden, Dresden,

Germany.

44 J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February

reading the diameter d in the center of the surround area,

as shown in Fig. 1. This approach considers the cone

moving as a rigid body and assumes a linear decay of the

displacement,

jx ðx; rÞjjx coilðxÞj

¼1; 0 � r � di

2

do � 2r

do � di

;di

2� r � do

2

8>>><>>>:

ð1Þ

versus the radius r in the surround area. Small showed

that the diameter d referred to the half-width of the

surround is a useful approximation of the effective

radiation area,

SD ¼p4

d2

i þ ðdo � diÞdi þðdo � diÞ2

3

" #’

p4

di þ do

2

� �2

¼ p4

d2 ð2Þ

as long as the surround is not wider than 10% of the outer

diameter do and the assumption in Eq. (1) is valid (see [5,

appendix]).

Significantly higher errors are found in woofers that use

a larger surround designed for realizing a high peak

displacement Xmax and especially in tweeters, head-

phones, horn compression drivers, and microspeakers,

where the outer part of the diaphragm itself is used as the

suspension system.

In those cases the displacement rises from zero at the

outer rim to the maximum value at the voice-coil position

by a function that is usually nonlinear (see solid line in

Fig. 2(b) and [5, fig. 8]) and depends on the particular

clamping condition of the rim, the thickness and curvature

of the diaphragm profile, and the material properties.

Theoretical estimates of the effective radiation area SD for

clamped plates, flat circular membranes, and other simple

geometries are given in [6] but may not be applicable to

real transducers having a much more complicated

geometry. Thus a precise value for the effective radiation

area SD has to be calculated from the total volume

velocity q(x) measured by using mechanical and/or

acoustical sensors [7]–[9].

This engineering report tackles the problem and

searches for simple, robust, and convenient measurement

techniques giving more reliable values of SD. At the

beginning the effective radiation area SD will be defined

by using the sound pressure output predicted by the

Rayleigh equation. In the second part the pros and cons of

the traditional techniques are discussed. A new technique

is presented which calculates SD from vibration and

radiator geometry using the results of modern laser

scanning techniques. The accuracy of the new method is

investigated and compared with traditional methods.

0.1 Symbols

ra far-field observation point

rc point on radiator surface

r0 distance between radiator center and observation

point ra

x angular frequency

x0 angular resonance frequency

p(x,ra) complex sound pressure at far-field observation

point

pbox(x) complex sound pressure in sealed test enclosure

x(x,rc) complex displacement at radiator surface

xcoil(x) complex voice-coil displacement

SD(x) complex value of effective radiation area de-

pending on frequency

SD effective radiation area describing piston mode at

resonance

q0 density of air

Sc total radiator surface

k wavenumber

ravg average radius (usually at voice-coil position)

rr outer radiator radius

q(x) total volume velocity generated by radiator

Cab acoustic compliance of air volume in sealed

enclosure

DVbox variation of air volume in test enclosure

Vbox total air volume of test enclosure

p0 static air pressure

c speed of sound

xcoilþDV(x0) voice-coil displacement at resonance fre-

quency measured in box with enlarged air

volume

pboxþDV(x0) sound pressure at resonance frequency

measured in box with enlarged air volume

Fig. 2. Variation of displacement with radius. (a) Geometry

of diaphragm and voice coil and graphically enhanced peak

and bottom displacement. (b) Amplitude of headphone

diaphragm displacement (—) measured at 297.4 Hz by laser

scanning compared to linear approximation (- - -) as used in

geometrical method for predicting SD.

J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February 45

ENGINEERING REPORTS MEASUREMENT OF EFFECTIVE RADIATION AREA

di inner diameter of surround area

do outer diameter of surround area

d diameter referred to center of surround area

1 DEFINITION OF SD

A radiator mounted in an infinite baffle generates a

sound pressure

p ðx; raÞ ¼ �x2q0

2p

ZSC

e�jkjra�rcj

jra � rcjx ðx; rcÞ dS ð3Þ

at a listening point ra in the far field using the Rayleigh

integral of complex displacement x(x,rc) over all points rc

on the surface Sc of the radiator, as illustrated in Fig. 3.

This approximation implies that the radiating surface has

a relatively flat shape and all dimensions are much

smaller than the wavelength of the sound considered for

calculating SD.

If the listening point ra is on axis, far away from the

radiator, and much larger than the diameter 2rr of the

radiator, then the distance jra – rcj ’ r0 may be assumed

constant and Eq. (3) simplifies to

p ðx; raÞ ¼ �x2q0

2pe�jkr0

r0

ZSC

x ðx; rcÞ dS: ð4Þ

The same sound pressure

p ðx; raÞ ¼jxq0

2pq ðxÞ e

�jkr0

r0

ð5Þ

is generated by a volume velocity

q ðxÞ ¼ jxZSC

x ðx; rcÞ dS ð6Þ

emitted by a point source in the center of the radiator. An

equivalent volume velocity

q ðxÞ ¼ jxx coilðxÞS DðxÞ ð7Þ

can be generated by a piston vibrating with the reference

displacement xcoil(x) and an effective radiation area

SD(x).

Combining Eqs. (6) and (7) gives the effective

radiation area,

S DðxÞ ¼q ðxÞ

jxx coilðxÞ¼

ZSC

x ðx; rcÞ dS

x coilðxÞð8Þ

which is a complex quantity and depends on frequency.

At low frequencies, where all parts of the radiator

vibrate in phase, the complex function SD(x) becomes a

real value, and the effective radiation area can be

defined as

SD ¼ jS Dðx0Þj ð9Þ

using the fundamental resonance frequency x0 as a

useful reference.

2 MEASUREMENT TECHNIQUES

The following dynamic measurement techniques make

no assumption regarding the displacement x(x,rc) on the

radiator surface.

2.1 Far-Field Measurement

Using Eqs. (5) and (7) the frequency-dependent

function becomes

S DðxÞ ¼ �2p

x2q0

r0

e�jkr0

p ðx; raÞx coilðxÞ

ð10Þ

which in turn becomes the effective radiation area,

SD ¼ jS Dðx0Þj ¼2pr0

x20q0

jp ðx0; raÞjjx coilðx0Þj

ð11Þ

at resonance frequency, according to Eq. (9). This

measurement technique requires a calibrated microphone

for measuring the sound pressure p(x,ra) on axis at a

distance ra from the radiator mounted in a sufficiently

large baffle and providing anechoic conditions. The

measurement of microspeakers, tweeters, and other

transducers having a high resonance frequency can be

accomplished by using a baffle according to IEC 60268-5

and medium-size anechoic rooms. This method is less

practical for measuring woofers in anechoic rooms, where

standing waves affect the accuracy of sound pressure

measurements below 100 Hz.

2.2 Near-Field Measurement in Baffle

Moreno et al. [5] suggested a measurement of the

sound pressure p(x,ra ’ rc) close to the surface of the

radiator to relax the requirements put on the acoustic

environment and calculate SD using the equation

SD ¼ jS Dðx0Þj ¼p4

2

x20q0

jp ðx0; raÞjjx coilðx0Þj

� �2

: ð12Þ

However, Moreno et al. [5] observed a significant

influence of the radiator geometry and measured a

Fig. 3. Sound pressure generated on axis at observation point

ra by radiator mounted in an infinite baffle.


KLIPPEL AND SCHLECHTER ENGINEERING REPORTS

difference of more than 10% in the SD value for flat and

conical pistons of the same size.

2.3 Box Method

A radiator mounted on a small enclosure, as shown in

Fig. 4, generates a sound pressure inside the enclosure

pboxðxÞ ¼

q ðxÞjxCab

ð13Þ

which at low frequencies is almost independent of the

microphone position [9]. Calculating the acoustic com-

pliance of the enclosed air,

Cab ¼Vbox

q0c2ð14Þ

by using the air volume Vbox, the air density q0, and the

speed of sound c gives the equivalent radiation area,

SD ¼ jS Dðx0Þj ¼Vbox

q0c2

jpboxðx0Þj

jx coilðx0Þj: ð15Þ

The volume Vbox of the air in the test enclosure has to

be measured carefully considering the free volume of the

enclosure and the air displaced by the loudspeaker parts.

In practice determination of Vbox limits the accuracy of

the measurement technique.

2.4 Differential-Box Method

Performing two measurements with different air

volumes, as illustrated in Fig. 5, dispenses with the

measurement of the static air volume displaced by the

transducers. Combining Eq. (15) of the first measurement

with

SD ¼Vbox þ DVbox

q0c2

jpboxþDV

ðx0Þjjx coilþDVðx0Þj

ð16Þ

of the second measurement and substituting Vbox in both

equations gives

SD ¼1

q0c2

DVbox

jx coilþDVðx0Þjjp

boxþDVðx0Þj

� jx coilðx0Þjjp

boxðx0Þj

: ð17Þ

A medical syringe, as shown in Fig. 6, may be used as

test enclosure for microspeakers, headphones, and tweet-

ers, which allows precise variations of the box volume

DVbox. Although the differential box puts the lowest

requirement on the acoustic environment, the major

disadvantage of this technique is that a minor leakage

may cause significant error in the measurement. Some-

Fig. 4. Measurement of internal pressure generated in a

sealed enclosure.

Fig. 5. Performing two measurements while using two test

enclosures having precise differences in air volume DVbox.

(a) First measurement. (b) Second measurement.

Fig. 6. Differential measurement of effective radiation area

of a headphone by using a syringe.



times there are also minor leakages in the material of the

diaphragm and in the connection to the surround and dust

cap, which affect measurement accuracy.

2.5 Laser Technique

A different approach to determine SD can be based on

the relations shown by Eqs. (8) and (9). There SD is

defined as the integrated displacement of the radiator

surface divided by the voice-coil excursion xcoil(x).

Replacing the integral by a sum over discrete elements

gives a good approximation of the effective radiation

area

S DðxÞ ¼X

x ðx; rc;iÞ � DSc;i

x coilðxÞð18Þ

where x(x,rc,i) is the displacement in the z direction and

DSc,i is the respective surface area projected on the r, uplane, as illustrated in Figs. 7 and 8. The vibration and

geometry of the radiator surface can be measured with

high accuracy by using a triangulation sensor and an

automatic scanning technique [10].

For the measurement of the coil excursion xcoil(x) it is

advisable not only to measure a single point but rather to

conduct spatial averaging by measuring multiple points,

for example, on a circle close to the voice coil,

x coilðxÞ ¼

Z2p

0

x ðx; ravg;uÞ du

2p: ð19Þ

The averaged radius ravg should be selected close to the

voice coil to be representative of the coil motion in the

piston mode. An optimal value for postprocessing of the

scanned data can be found using the graphical user

interface depicted in Fig. 9.

2.5.1 Frequency Dependence of SD(x)

By applying Eq. (18) a frequency-dependent effective

radiation area SD(x) is obtained. At low frequencies

below the breakup of the cone the amplitude of SD(x) is

constant and represents the effective radiation area of the

piston motion, which is equal to the T/S parameter SD

[compare Eq. (9)].

The graphs of jSD(x)j are shown in Fig. 10 for a set of

different ravg values in the center of an oval microspeaker.

Over a broad frequency range of 10 Hz to 1 kHz the

derived SD is nearly constant. Averaging in that frequency

range will further improve the accuracy of the nominal SD

of the driver. At higher frequencies SD(x) is no longer

constant but depends on the specific vibration properties

at the averaged radius ravg in relation to the total volume

flow q(x).

As long as the ravg used is close to the voice coil of the

driver, the respective SD(x) represents a linear transfer

function between a given coil displacement xcoil(x) and

the far-field sound pressure on axis. See Eq. (20), which

can be derived from Eq. (10),

p ðx; r0Þ ¼ �x2q0

2pe�jkr0

r0

S DðxÞx coilðxÞ: ð20Þ

2.5.2 Average Radius ravg

If ravg is too far from the voice coil then the coil

excursion xcoil(x) will contain an error, which leads to an

incorrect SD. The measured values of xcoil and SD are

shown in Fig. 11 for various locations of the averaged

radius ravg and for a frequency of 100 Hz. It can be seen

that the amplitude of the coil displacement is nearly

constant in the center of the driver up to a radius of 7 mm

and gives a valid value of SD ¼ 3.4 cm2. If the average

radius ravg is larger than the displacement, xcoil drops to

zero until the outer driver edge and the derived SD goes to

infinity.

Fig. 7. Laser scanning technique for measuring vibration and

geometry of radiator surface.

Fig. 8. Cone surface elements DSc,i.



2.5.3 Measurement Grid Resolution

The resolution of the scanning grid for measuring the

displacement x(x,rc,i) is important for the accuracy of SD.

An investigation of the effect of reducing the number of

radial and angular measurement points was conducted for

the two microspeakers shown in Fig. 12. In both cases a

full, detailed scan of 3201 points was coarsened by re-

moving the data from single measurement radii or angles.

The average radius has been set to 6 mm for the circular

speaker and to 10 mm for the oval speaker.

The nominal value of SD remains very accurate for the

circular speaker, even when only three radial lines or four

angular lines were kept. Even a combination of three

radial and four angular steps will result in SD¼3.375 cm2,

which is only 1% less than the full resolution result of SD

¼ 3.408 cm2 (see Fig. 13). This means that in this case,

with x(x,rc,i) measured at only 9 points, it is possible to

determine the SD value of the driver with acceptable

accuracy. But it should be stated that this result cannot be

applied to all circular speakers. In general at least a higher

radial resolution is necessary to cover the decay of the

displacement amplitude toward larger radii properly. As

long as the driver shape is axis symmetrical, a higher

angular resolution will only improve the signal-to-noise

ratio of the measurement and can help with evaluating the

influence of rocking modes correctly.

For the oval speaker the error becomes larger if too few

radial or angular steps are considered. In particular a

Fig. 9. User interface of SD calculation tool required to specify voice-coil position.

Fig. 10. Effective radiation area jSD(x)j versus frequency

using eight different values of ravg for averaging voice-coil

displacement.

Fig. 11. (a) Cross-sectional view of radiator with enhanced

peak and bottom displacement. (b) Mean value of displace-

ment measured at a circle of radius ravg on diaphragm. (c)

Effective radiation area SD versus averaged radius ravg at

100 Hz.



higher angular resolution is necessary here to measure the

displacement distribution on an asymmetrical speaker

with sufficient accuracy (see Fig. 14).

The measured cone displacement data should cover the

complete driver area to give a proper value for SD by

applying Eq. (18). But sometimes this is not possible as

when certain cone regions are not accessible by laser

because they are hidden behind a cover. Then the

integrated surface displacement will be too low, which

leads to a too low SD value.

Furthermore the triangulation laser can produce

errors when measuring the vibration at sharp edges or

poorly reflective spots on the cone. These erroneous

measurement points will also affect the calculation of

SD. The scanning software used for the present

examples can recognize erroneous points by correlat-

ing two subsequent measurements from different

distances. These points will be removed from all

further analyses by setting the measured displacement

to zero. Again the integrated surface displacement will

be lower depending on the number and location of the

erroneous points.

Fig. 12. Intensity plot of cone vibration and sectional view. (a) Circular microspeaker. (b) Oval microspeaker.

Fig. 13. Calculated values of effective radiation area SD for

circular speaker. (a) Radial grid resolution. (b) Angular

resolution.

Fig. 14. Calculated values of effective radiation area SD for

oval speaker. (a) Radial grid resolution. (b) Angular

resolution.



3 DISCUSSION

The laser technique proposed in Section 2.5 performs a

vibration measurement of the complete cone. Measuring

the displacement at several points instead of only one

point is also advisable for the single-box technique

(Section 2.3) and the differential-box technique (Section

2.4). The reason is that especially small speakers without

properly centering suspension are susceptible to asym-

metrical vibrations and rocking modes, as shown in Fig.

15.

These vibration modes usually occur at low frequencies

where the speaker is supposed to perform a piston motion.

The displacement measured at only one point will not be

representative of the motion of the complete cone in the

presence of asymmetrical vibration modes. Therefore it is

necessary to perform a set of vibration measurements at

selected points on the cone and average the measured

data.

4 CONCLUSION

An accurate measurement of the effective radiation

area SD is the basis for a reliable prediction of acoustical

output. A 12% error in SD causes an error of 1 dB in

passband sensitivity.

Exploiting the geometry of the radiator by reading

only the diameter in the center of the surround is the

simplest technique, which may be applicable to woofers

where the width of the surround is small compared to

the total diameter of the radiator and the total volume

velocity is generated by the cone vibrating as a rigid

body. This technique is not reliable for headphones,

microspeakers, compression drives, and other transduc-

ers where the vibrations vary significantly with the

radiator surface. Here the total volume velocity has to be

measured using acoustical and/or optical sensors. The

far-field measurement suggested in this engineering

report is a simple technique giving more accurate results

than near-field measurements but it requires a large

baffle and an anechoic environment. The differential-

box technique puts lower requirements on the test

equipment but requires careful sealing of the test

enclosure.

Measurement of the voice-coil displacement requires a

mechanical measurement, which can be accomplished by

optical laser triangulation sensors. The measurement of

multiple points is required to consider non-axis symmet-

rical modes, as found on most transducers having no

spider.

If the radiator surface is completely accessible to an

optical laser scanner no acoustical measurements are

required and a precise value of the effective radiation area

can be calculated from the measured vibration and

geometry.

5 REFERENCES

[1] L. Beranek, Acoustics (Acoustical Society of

America, New York, 1996), p. 231.

[2] R. H. Small, ‘‘Direct-Radiator Loudspeaker System

Analysis,’’ J. Audio Eng. Soc., vol. 20, pp. 383–395 (1972

June).

[3] D. Clark, ‘‘Precision Measurement of Loudspeaker

Parameters,’’ J. Audio Eng. Soc., vol. 45, pp. 129–141

(1997 Mar).

[4] W. Klippel and U. Seidel, ‘‘Fast and Accurate

Measurement of Linear Transducer Parameters,’’ present-

ed at the 110th Convention of the Audio Engineering

Society, J. Audio Eng. Soc. (Abstracts), vol. 49, p. 526

(2001 June), convention paper 5308.

[5] J. N. Moreno, R. A. Moscoso, and S. Jonsson,

‘‘Measurement of the Effective Radiating Surface Area of

a Loudspeaker Using a Laser Velocity Transducer and a

Microphone,’’ presented at the 96th Convention of the

Audio Engineering Society, J. Audio Eng. Soc. (Ab-

stracts), vol. 42, p. 405 (1994 May), preprint 3861.

[6] R. Ballas, G. Pfeifer, and R. Werthschutzky,

Eletromechanische Systeme der Mikrotechnik und Me-

chatronik (Springer, Berlin Heidelberg, 2009).

[7] S. Jonsson, J. N. Moreno, and H. Bog, ‘‘Measure-

ment of Closed Box Loudspeaker System Parameters

Using a Laser Velocity Transducer and an Audio

Analyzer,’’ presented at the 92nd Convention of the

Audio Engineering Society, J. Audio Eng. Soc. (Ab-

stracts), vol. 40, p. 433 (1992 May), preprint 3325.

[8] J. N. Moreno, ‘‘Measurement of Loudspeaker

Parameters Using a Laser Velocity Transducer and

Two-Channel FFT Analysis,’’ J. Audio Eng. Soc., vol.

39, pp. 243–249 (1991 Apr).

[9] D. K Anthony and S. J. Elliot, ‘‘A Comparison of

Three Methods of Measuring the Volume Velocity of an

Acoustic Source,’’ J. Audio Eng. Soc., vol. 39, pp. 355–

366 (1991 May).

Fig. 15. Irregular vibration of headphone diaphragm. (a)

Asymmetric motion. (b) Rocking motion.



[10] W. Klippel and J. Schlechter, ‘‘Measurement

and Visualization of Loudspeaker Cone Vibration,’’presented at the 121st Convention of the Audio

Engineering Society, J. Audio Eng. Soc. (Abstracts),

vol. 54, pp. 1251, 1252 (2006 Dec.), convention paper

6882.

THE AUTHORS

W. Klippel J. Schlechter

Wolfgang Klippel studied electrical engineering at the

University of Technology, Dresden, Germany. After

graduating in speech recognition, he joined VEB

Nachrichtenelektronik in Leipzig, Germany, where he

was engaged in the research of transducer modeling,

acoustic measurements, and psychoacoustics. In 1987 he

received a Ph.D. degree in technical acoustics. After

spending a postdoctoral year at the Audio Research

Group in Waterloo, Ont., Canada, and working at Harman

International, Northridge, CA, he returned to Dresden in

1997 and founded Klippel GmbH, which develops novel

kinds of control and measurement systems dedicated to

loudspeakers and other transducers. In 2007 he became

professor of electroacoustics at the University of

Dresden.

Joachim Schlechter was born in Oschatz, Germany, in

1981. He studied computer and media science with a

focus on acoustics and signal processing at the University

of Technology, Dresden, Germany. He attended a master

program for sound and vibration at the Chalmers

Technical University, Gothenburg, Sweden, from which

he received a Master’s degree in the field of active noise

control in 2005. He also received a Diploma degree in

computer science, Dresden, Germany, in 2006. The

supervision of his thesis, ‘‘Visualization of Vibrations of

Loudspeaker Membranes,’’ was shared between the

University of Technology of Dresden and Klippel GmbH.

After graduating he joined Klippel GmbH, where he is

currently engaged in the research and development of

loudspeaker control and measurement systems.



JAES_V59_1_2_PG44hirez

Documents

Transcript of JAES_V59_1_2_PG44hirez