Acoustics II: loudspeakers

loudspeakers

transducerprinciples

electrodynamic

electrostatic

radiation

directivity

radiated power

velocity distribution on amembrane

electrodynamicloudspeaker

equivalent network

cabinets

electrical impedance

sound pressure frequencyresponse

Thiele-Small parameters

cabinet types

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magnetostaticloudspeaker

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Acoustics II:loudspeakers

Kurt Heutschi2013-01-18

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loudspeakers: transducerprinciples

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transducer principles

membrane driver principles:

I electrodynamic

I electrostatic

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electrodynamic transducer

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electrodynamic transducer

drive mechanism: force F acting on a wire of length `carrying current I and located in a magnetic field B :

F = B × ` · I

the other way round, a voltage U is induced for a wiremoving with velocity u:

U = B × ` · u

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electrostatic transducer

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electrostatic transducer

drive mechanism: electrostatic force F acting on a platecondenser of area S and distance x and for a voltage U :

F =ε0SU

2

2x2

with: ε0: electric field constant = 8.85×10−12 AsV−1m−1

I non-linear relation between force and voltage makesbiasing necessary!

I ≈ linear behavior for small variations

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sound radiation by a movingmembrane

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sound radiation by a moving membrane

diffraction of plane wave at a circular opening (assumingKirchhoff approximation):

movie: diffraction at circular opening

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directivity

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directivity

Calculation of sound pressure p̌ for a piston

I of area S

I with velocity v̌nI in an ∞ wall

with help of the Rayleigh-Integral:

p̌ =jωρ

2π

∫S

v̌n1

re−jkrdS

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Far field directivity for circular piston

in larger distances: 1/r term can be assumed constant(= r0)

p̌(φ) ≈ v̌nr0jka2ρc

J1(ka sin θ)

ka sin θ

wherev̌n: velocity of the pistonr0: reference distancek : wave number = 2π/λa: radius of pistonJ1: Bessel functionθ: angle to the receiver position

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Directivity

Directivity of the piston radiator in the ∞-wall:

ka = 1 ka = 2 ka = 5 ka = 10

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radiated power

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radiated power

Radiation of a membrane depends on:

I membrane velocity (usually piston movement isassumed)

I membrane area

I surrounding of the membrane

I medium

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Radiated power

radiation impedance ZRo :

ZRo =p

v

wherep: average sound pressure at the surface of themembranev : velocity of the membrane (normal component)

I describes loading of the membrane

I calculation with wave theoretical methods

I often alternative definition with volume flow is used:ZR = p

Q

I radiation impedance is complex quantity → activeand reactive component

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Radiated power

effectively radiated power W by moving membrane is

W = Q2Re[ZR ]

withQ: volume flow (product of velocity and membrane area)Re[ZR ]: real (active) part of radiation impedance

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Radiated power

examples of radiation impedances (rel. to ρc):

piston in ∞-wall free piston

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Radiated power

simulation of radiation impedances in equivalentnetworks (frequency range with f 2 dependency):

I often series arrangement of frequency dependentresistance (real part) and inductance (imaginarypart)

I approximation of the load in networks: parallelarrangement

I resistance R = ρc (dominates above ka = 1)I inductance L = ρa/

√2 (dominates below ka = 1)

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velocity distribution on a membraneI investigation of the membrane movement of a 20

cm woofer

I measurement method: Laser vibrometer (Doppler)

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velocity distribution on a membrane

frequency: 218 Hzmovie: frequency: 218 Hz

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Velocity distribution on a membrane


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Velocity distribution on a membrane


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electrodynamic loudspeaker

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electrodynamic loudspeaker:principle of operation

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principle of operation

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Electrodynamic loudspeaker:Equivalent network

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Equivalent networkelements to consider:

MAR ,RAR acoustical mass and resistance of the air atthe rear side of the membranecorresponding to real and imaginary part ofthe radiation impedance

m, s mass of the membrane and the coil,stiffness of the membrane

Rm mechanical friction (suspension of themembrane)

MAV ,RAV acoustical mass and resistance of the air atthe front side of the membranecorresponding to real and imaginary part ofthe radiation impedance

RE , LE electrical resistance and inductance of thecoil

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Equivalent network

I B × `: force factor

I electrical interface:I source current → force FI membrane velocity → induced voltage

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Equivalent network

elimination of the gyrators by dual conversion withr = 1/A, omit the 1:1 transformers

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Equivalent network

behavior of volume flow Q:

I low frequencies: capacitor dominates → Q ∼ ω

I resonance frequency: Q limited by resistances

I high frequencies: inductances dominate → Q ∼ 1/ω

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Equivalent network

I radiated sound power W = Q2 · Re[ZR ]

I independent of frequency for Q ∼ 1/ω andRe[ZR ] ∼ ω2

I → operation above resonance

I → piston mounted in ∞-wall

I → upper limiting frequency: ka < 1

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Electrodynamic loudspeaker:Nonlinearities

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Nonlinearities

I for large amplitudes, the element values will varyaccording to the membrane position

I → nonlinear behavior

I critical:I stiffness of the outer and inner suspensionI inductance of the moving coil LEI force factor B × `

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electrodynamic loudspeaker:mounting in a cabinet

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mounting in a cabinetI Chassis in ∞-wall realized by mounting in a cabinetI alterations:

I no radiation impedance at rear sideI increasing stiffness due to enclosed air in the

cabinet → acoustical compliance CA

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Mounting in a cabinet

consequences:

I total capacitance of series resonance circuit islowered

I increased resonance frequency

I the smaller the volume, the higher the resonancefrequency

I → cabinet volume not too small and:I filling with porous material → isothermal

behaviour → effective volume increased by 15%

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Mounting in a cabinetequivalent network so far:

further steps:

I combination of elements of the same type

I dual conversion of the complete network withr = 1/A

I ignore real part of the radiation impedance (o.k. forlow frequencies)

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quantity of interest: volume flow Q appears as voltageU ′S :

U ′S =Q

A

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elimination of transformer by impedance scaling with(B × `)2

quantity of interest: volume flow Q is transformed intovoltage US :

US = (B × `)U ′S = (B × `)QA

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I potential problems introduced by the cabinet(compared to the ∞-wall):

I mechanical vibrations of the cabinet → boomingI diffraction at the edgesI standing waves (resonances) in the cabinet

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electrodynamic loudspeaker:electrical impedance

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Electrical impedance

?

I characteristic impedance (typical values: 4 or 8Ohm)

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electrodynamic loudspeaker:sound pressure frequency

response

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sound pressure frequency response

I restriction to low frequenciesI → omnidirectional radiation

p(d) =

√W ρ0c

4πd2=

Q√

Re[ZR ]ρ0c√4πd

wherep(d): sound pressure in distance dW : radiated sound powerQ: volume flow of the membrane (velocity times area)Re[ZR ]: Real part of the radiation impedance

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Sound pressure frequency response

radiation impedance (approximation for low frequencies,mounted in a cabinet):

Re[ZR ] =ρ0c

2(ka)2

1

a2π=ρ0ω

2

2πc

withk : wave numbera: radius of the membraneω: angular frequency

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determination of Q: US = (B × `)QA

neglecting LE :

US

U≈ jωLRRR

−ω2LRRRCRRE + jω(LRRR + LRRE ) + RRRE

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p(d)

U=

ρ0a2

d√

8B`

1

j

1

RECR

−ω2LRCR

−ω2LRCR + jω LRRR+LRRE

RRRE+ 1

frequency dependency → high-pass function 2. order:

G (jω) =−ω2T 2

c

−ω2T 2c + jω Tc

QTC+ 1

withTc = 1

ωc

ωc : lower limiting frequency of the high-pass filterQTC : quality factor of the high-pass filter

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p(d)/U for different values of QTC :

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electrodynamic loudspeaker:Thiele-Small parameters

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Thiele-Small parameters for the characterization of afree chassis:

resonance frequency fs in Hz : lower frequency end ofthe range of usage

compliance equivalent volume VAS in l stiffness of themembrane and suspension, expressed asequivalent air volume of equal stiffness

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Thiele-Small-Parameters

mechanical Q factor QMS mechanical damping of theresonance (by RR)

electrical Q factor QES : electrical damping of theresonance (by RE and possible outputresistance of voltage source)

total Q factor QTS QTS = QMSQES

QMS+QES

membrane area A in m2

DC-resistance RE in Ohm

force factor B × ` in T×m

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Thiele-Small-Parameters

I determination by two electrical impedancemeasurements

I case a): chassis free

I case b): chassis mounted in a cabinet oralternatively loaded with additional mass put on themembrane

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Thiele-Small-Parameters: mounted in a

cabinet

free chassis specified by fS , QTS , VAS

behavior of chassis mounted in a cabinet of volume VB :

fc = fs

√VAS

VB+ 1

QTC = QTS

√VAS

VB+ 1

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Thiele-Small-Parameters: mounted in a

cabinetexample → amplitude responses for different cabinetvolumes:

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Electrodynamic loudspeaker:velocity and displacement of

the membrane

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Velocity and displacement of the

membrane

I in the range of operation: membrane velocity v ∼ 1f

I → membrane displacement: x ∼ 1f 2

I → maximal excursion at lower frequency end ofrange of operation

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electrodynamic loudspeaker:bass reflex cabinet

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bass reflex cabinetI extension of frequency range of operation to lower

frequenciesI strategy: introduction of additional resonance:

I acoustical compliance: air volume in cabinet: CA

I acoustical mass: cylinder of air in tube: Mp

I around resonance strongest oscillations of aircolumn

I → relevant sound radiation

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Bass reflex cabinet

Spice simulation:chassis: fs = 31 Hz; QTS = 0.3; VAS = 118 l

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Bass reflex cabinet

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Bass reflex cabinet

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Bass reflex cabinet

possible problems:

I not satisfying transient behavior

I possible flow noise at end of tubeI overcome by passive membrane systemsI realization of mass by mechanical means

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Electrodynamic loudspeaker:Bandpass cabinet

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Bandpass cabinet

I sub-woofer with acoustical low-pass filtering

I bandpass cabinet with chassis mounted in theinterior of the cabinet

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electrodynamic loudspeaker:feed-back

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feed-back

I aim of feed-back systems:I controlling oscillation at resonanceI lowering of non-linearities (large membrane

excursions)

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Feed-back

I observation of membrane condition:I measurement of membrane velocity by additional

moving coilI measurement of membrane position by capacitive

sensorI measurement of membrane position by optical

means

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Electrodynamic loudspeaker:Control of membrane

velocity

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Control of membrane velocity

I neutralization of driving coil DC resistanceI powering by amplifier with negative output

resistance

I → direct control of membrane velocity US ≈ U

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Control of membrane velocity

I excellent transient behavior

I electrical 1/f -correction necessary to fulfill Q ∼ 1/f

I control effect gets lower for high frequencies as LEbecomes important

I caution: RE − RA > 0 → stability condition!

I |RE | ≈ |RA| is very delicate (temperaturedependency of RE )

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Control of membrane velocityrealization example: Studer A723

I negative output resistance:I increase in output current → increase in output

voltage

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Electrodynamic loudspeaker:Digital Equalizing

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Digital Equalizing

I impulse response of loudspeaker: h(t)

I inverse filter g(t) where g(t) ∗ h(t) = δ(t)

I pre-filtering of audio signal with g(t)

I but:I physical limitations (1/0→∞)I h(t) depends on radiation angleI h(t) depends on membrane excursion

(nonlinearities)

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horn loudspeaker

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horn loudspeakerI improved impedance matching (membrane

movement → air)I significantly higher efficiencyI more pronounced directivity

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horn loudspeaker

common horn shape: exponential function:

S(x) = STemx

whereS(x): cross sectional area at xST : cross sectional area at the throat (x = 0)m: geometrical constant

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horn loudspeaker

wave equation for the exponential horn:

∂2p

∂t2−c2m∂p

∂x− c2

∂2p

∂x2= 0

guess for p(t):

p(t) = p̂eaxe jωt

with coefficient a:

a = −

(m

2+ j

√4(ω/c)2 −m2

2

)

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Horn loudspeaker

determination of sound particle velocity:relation between p and v :

∂p

∂x= −ρ∂vx

∂t

in complex writing:

v = − 1

jωρ

∂p

∂x

with p inserted:

v =1

jωρ

(m

2+ j

√4(ω/c)2 −m2

2

)p

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Horn loudspeaker

radiation impedance:

ZRo =p

v=

2jωρ

m + j√

4(ω/c)2 −m2

discussion of ZRo :

2(ω/c) < m ZRo is purely imaginary

2(ω/c) ≥ m ZRo has a real and imaginary part with

Re[ ZRo ] = ρc√

1− m2

4(ω/c)2

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Horn loudspeakerexemplarily radiation impedance for m = 3.7:

I region of usage: constant impedanceI → volume flow Q has to be constant (operation at

resonance with strong damping)I caution: non-linear distortions due to high flow

amplitudes in the throat

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Horn loudspeaker

I practical design rules (reflections at the end can beignored):

I length of the horn > 3λI circumference at the end > λ

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Horn loudspeaker

compression driver (principle):

I acoustical transformer:I low velocity of large membrane → high velocity in

small throat

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Horn loudspeakercompression driver (practical design):

I phase plugs:I suppression of cavity resonancesI guarantee in-phase superposition of velocities in

throat cross section

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electrostatic loudspeaker

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electrostatic loudspeakerI push-pull principle → sandwich structure:

I two perforated stationary electrodes (acousticallytransparent)

I lightweight stretched membrane in between (0.2g/dm2)

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electrostatic loudspeaker

I bias voltage needed

I excellent transient behavior (lightweight andhomogeneously driven)

I only small excursions possible → large membraneareas needed

I in tendency weak for bass frequencies

I focusing of high frequencies

I reflectors at rear side of speaker may cause problems

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magnetostatic loudspeaker

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magnetostatic loudspeaker

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loudspeaker systems

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loudspeaker systems

I radiation of the whole audio frequency range (20 Hz. . . 20 kHz) with one chassis is problematic as

I radiation at low frequencies requires largemembrane excursion or large membrane areas →low upper limiting frequency (kr = 2)

I Doppler distortions (frequency modulations of highfrequency signal components by low frequencies

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Loudspeaker systems

I solution:I subdivision of whole frequency range into several

bands by frequency dividing network and radiationby different specialized chassis

I typical 2 and 3 way systems

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Loudspeaker systems

potential difficulties:

I destructive interferences in the transition region dueto:

I differences in phase response of the chassisI differences in path length: chassis → listener

solutions:

I adjustment of phase response

I adjustment for equal path lengths by geometricalmeans

I optimal: concentric dual chassis

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Loudspeaker systems

power handling:

I defined by:I max. membrane excursion → specified by peak

powerI heating → specified by average power

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Loudspeaker systems

continuous power testing with standard third-octavespectrum according to IEC 268:

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Loudspeaker systems

cumulated power for the spectrum according to IEC 268:

caution: maximum power handling of a chassis isspecified as power of the whole system!

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loudspeaker measurements

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loudspeaker measurements

I frequency response of the electrical impedance atthe terminals

I frequency response of sound pressure on axis

I impulse response of sound pressure on axis

I cumulative decay spectrum of sound pressure onaxis

I directivity of sound pressure for various discretefrequencies

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cumulative decay spectrum of sound

pressure

I description of the transient behavior as a functionof frequency

I experiment: excitation with tone burst signal

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cumulative decay spectrum

I post-oscillation of the membrane produces soundpressure signal sω(t)

sω(t) = Hω(t) sin(ωt)

withHω(t): modulation of amplitudeω: angular frequency of the excitation signal

→ waterfall diagram of Hω(t)

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efficient generation by taking FFT of time-windowedimpulse response:

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cumulative decay spectrumexample of a measurement of a mid-range speaker:

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exotic transducers

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NXT - distributed mode -loudspeaker

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Distributed mode - loudspeaker

history:

I 1991 first patent by Ken Heron of Britain’s DefenceEvaluation and Research Agency (DERA)

I British company NXT gets a licence on the principleof operation.

I further investigation of the concept anddevelopment of the corresponding technology

I today, NXT sells licences to other manufacturers

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Distributed mode - loudspeaker: principle

I NXT speakers consists of large panels

I drivers excite as many partial oscillations as possible(bending waves) → panel oscillates with ’arbitrary’local velocity distribution

I local oscillation pattern leads to energeticsummation at a receiver → omnidirectionalradiation characteristics (figure-of-eight in case ofmounting in free space)

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construction:

I large panels z.B. 60x60 cm

I excitation by one or several drivers

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typical amplitude response:

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typical impulse response [ms units]:

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applications:

I invisible P.A. systems (e.g. speaker arrays mountedin a ceiling)

I in case of special requirements (fire retardant,cleanable)

I optical-acoustical double-usage (e.g. surface foroptical projection acts as speaker as well)

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Air motion transformer

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Air motion transformer

history:

I Oskar Heil, USA

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Air motion transformer: principle

I folded plastic foil between permanent magnet (largeair gap → strong magnet required

I conductors along the folds (up and down)

I force across the membrane surface → folds areopened and closed

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air motion transformer

advantages:

I small membrane movement → efficient volumevariation → high volume flow

I typical transformation ratio: 1:5

I excellent transient behavior

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impulse response eton:

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impulse response visaton tweeter:

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Acoustics II: loudspeakers