D. G. Meyer School Of Electrical & Computer Engineering

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Loudspeaker ParametersD. G. MeyerSchool of Electrical & ComputerEngineering

Outline Review of How Loudspeakers WorkSmall Signal Loudspeaker ParametersEffect of Loudspeaker CableSample LoudspeakerElectrical Power NeededSealed Box Design Example

How Loudspeakers Work

How Loudspeakers Are Made

Fundamental Small Signal Mechanical ParametersSd – projected area of driver diaphragm (m2)Mms – mass of diaphragm (kg)Cms – compliance of driver’s suspension (m/N)Rms – mechanical resistance of driver’s suspension(N s/m) Le – voice coil inductance (mH) Re – DC resistance of voice coil (Ω) Bl – product of magnetic field strength in voice coilgap and length of wire in magnetic field (T m)

Small Signal ParametersThese values can be determined by measuring the input impedance of thedriver, near the resonance frequency, at small input levels for which themechanical behavior of the driver is effectively linear. Fs – (free air) resonance frequency of driver (Hz)– frequency at which the combination of the energy stored inthe moving mass and suspension compliance is maximum,which results in maximum cone velocity– usually it is less efficient to produce output frequenciesbelow Fs– input signals significantly below Fs can result in largeexcursions– typical factory tolerance for Fs spec is 15%

Measurement of Loudspeaker Free-Air Resonance

Small Signal ParametersThese values can be determined by measuring the input impedance of thedriver, near the resonance frequency, at small input levels for which themechanical behavior of the driver is effectively linear. Qts – total Q of driver at Fs– unitless measurement, characterizing the combinedelectrical and mechanical damping of the driver– proportional to the energy stored, divided by the energydissipated– most drivers have Qts values between 0.2 and 0.5

Small Signal ParametersThese values can be determined by measuring the input impedance of thedriver, near the resonance frequency, at small input levels for which themechanical behavior of the driver is effectively linear. Qms – mechanical Q of driver at Fs– unitless measurement, characterizing the mechanicaldamping of the driver, i.e., losses in the suspension(surround and spider)– varies roughly between 0.5 and 10 (typical value is 3)– high Qms indicates lower mechanical losses– main effect of Qms is on impedance: high Qms drivers displaya higher impedance peak

Small Signal ParametersThese values can be determined by measuring the input impedance of thedriver, near the resonance frequency, at small input levels for which themechanical behavior of the driver is effectively linear. Qes – electrical Q of driver at Fs– a unitless measurement, describing the electrical dampingof the speaker– as the coil of wire moves through the magnetic field, itgenerates a current which opposes the motion of the coil(“back EMF”)– the back EMF decreases the total current through the coilnear Fs, reducing cone movement and increasing impedance– Qes is the dominant factor in voice coil damping for mostdrivers (depends on amplifier output impedance)

Aside: How Does Loudspeaker Cable And PowerAmplifier Output Impedance Affect Performance? Damping is a measure of a power amplifier's ability tocontrol the back EMF motion of the loudspeaker coneafter the signal disappears The damping factor of a system is the ratio of theloudspeaker's nominal impedance to the totalimpedance driving it Example: Amplifier with damping factor of 300 (bigger isbetter) driving an 8ΩΩ load means that the outputimpedance is 0.027ΩΩ (lower is better) Impedance of speaker cable used can significantlyreduce the damping factor (larger gauge wire has lowerimpedance)

Small Signal ParametersThese values can be determined by measuring the input impedance of thedriver, near the resonance frequency, at small input levels for which themechanical behavior of the driver is effectively linear. Vas – equivalent compliance volume (volume of airwhich, when acted upon by a piston of area Sd, hasthe same compliance as the driver’s suspension)– measure of the “stiffness” of the suspension with the drivermounted in free air– represents the volume of air that has the same stiffness asthe driver’s suspension when acted upon by a piston of thesame area (Sd) as the cone– larger values mean lower stiffness (and generally requirelarger enclosures)– Vas varies with the square of the speaker’s diameter– typical factory tolerance for Vas is 20-30%

Sample LoudspeakerSensitivity 86 dBFs 65 HzRe 6.6ΩLe 0.48 mHQts 0.365Qes 0.436Qms 2.25Vas 3.15 liters

Aside: How Loud Does This Thing Get?(And, How Much Power Do I Need?) Relating electrical power needed to producedesired SPL at a given listening distance:– sensitivity rating of loudspeaker (typically specas 1 m on-axis with input of 1 electrical watt)– acoustic level change/attenuation betweenloudspeaker and farthest listening position Example: want 90 dB program level at listeningdistance of 4 m outdoors (i.e., no reinforcement ofsound due to room reflections)– loudspeaker sensitivity measured as 86 dB– acoustic level change 20 log (4) 12 dB– SPL required at source is 90 12 102 dB– need 16 dB above 1 watt, or 10 (16/10) 40 W– check to make sure driver can handle it!

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Enclosure-Related Parameters EBP Fs / Qes– used loosely to decide what type of enclosure will be bestfor a given speaker (“rule of thumb”)– sealed box EBP 50– vented enclosure EBP 100 Fc sealed enclosure resonance Vc internal volume of a sealed enclosure Vb internal volume of a vented enclosure Fb bass reflex enclosure resonance Lv length of port Qtc – desired value determines size and type of enclosure– value of 0.707 is what most designers aim for (yieldsflattest response)– enclosures designed to enhance bass may range from 0.8to 1.1 (bigger value, “boomier” sound)– enclosures that are too big (Qtc 0.707) can sound “tinny”

Sealed Box

Sealed Box Theoretically an infinite baffle with an additionalstiffness component added (to the existingsuspension compliance) due to the springinessof the air volume trapped in the enclosure A smaller box will have a greater stiffnesscontribution than a larger one – in sealed boxsystems the air restoring force is normally madedominant (compared with that of the driversuspension)

Electromechanical Equivalent for Driver in aSealed BoxSimplified form –system response isthat of a dampedsingle-resonant circuit

Response Shape Critically damped (QTC 0.5): response is -6 dB atresonance (no ringing/overshoot in transient response) Butterworth alignment (QTC 0.7): response is -3 dB atresonance (response is “maximally flat” and has goodtransient behavior) QTC 1: provides greater bandwidth at expense oftransient response accuracy Chebychev alignment (QTC 1.1): 2 dB peak in responseat resonance – results in optimum efficiency alignment fora sealed box system, and is permissible for a small systemof limited bandwidth (e.g., FC of 65 Hz and above)

Behavior of a Closed-Box Loudspeaker System forSeveral Values of Total System QTCAmplitude vs. NormalizedFrequency ResponseNormalized Step Response

Enclosure Volume and Efficiency The maximum efficiency even a large sealed-boxenclosure can achieve is small (1-2%) Increasing the box cutoff frequency increases thetheoretical efficiency The closed-box system efficiency in the passband region(system reference efficiency) is the reference efficiency ofthe driver operating with the particular value of air-loadmass provided by the system enclosure Reference efficiency (η0 ) calculation:η0 4π2/c3 x (FS3 VAS)/QES 4π2/c3 x (FC3 VAT)/QECwhere VAT is a volume of air having the same total acousticcompliance as the driver suspension and enclosure actingtogether

Relationship of Maximum Reference Efficiency toCutoff Frequency and Enclosure VolumeExample: An 8 cu. ft. (e.g.2’x2’x2’) sealed box with areference efficiency (η0 ) of2% would have a cutofffrequency (Fc) of slightlyless than 40 HzChart for Chebychev Alignment

Box Filling or Damping Stuffing may offer an apparent air volume increase of upto 15% Additionally, stuffing may add a mass component due tophysical movement of the filling at lower frequencies Combined effects lower the system resonance and mustbe accounted for in the design (the effective cone massincrease could be as much as 20%) Very dense fillings will increase frictional air losses in theenclosed air volume and augment the damping If system is designed correctly, such damping is notrequired, but may help control a system where the QTC istoo high (e.g., due to inadequate magnet strength) Movement of the filling is undesirable!

Sealed Box Design Compliance Ratio: α VAS / VB System-driver relationships:QTC/QTS QEC/QES FC/FS (α 1)0.5FC / QTC FS / QTS where QTS is the total Q of thedriver at FS for zero source resistance, i.e.QTS QESQMS /(QES QMS) These equations show that for any enclosure-drivercombination the speaker resonance frequency mustalways be lower than that of the system resonancefrequency (i.e., value of α) and Q will be in the sameratio as those of the driver, but individually raised by afactor of (α 1)0.5 Provides guidance for both “fixed driver” designs anddesigns where only a final system specification is given

Ratio of Closed-Box System Resonance Frequency (FC) and Total Q (QTC)to Driver Resonance Frequency (FS ) and QT,as a Function of the System Compliance Ratio (α)fC box cutofffrequencyfS loudspeakerfree-air resonancefrequencyCompliance Ratio: α VAS / VB

Sealed Box Design Example FS 31 HzEBP 40 use sealed enclosureQES 0.77QMS 1.89VAS 65.8 litersη0 4π2/c3 x (FS3 VAS)/QES (9.64 x 10-7) x (313 x 65.8)/0.77 2.45%effective diaphragm radius 10.2 cm (0.102 m),giving SD 3.27 x 10-2 m2peak linear displacement (Xmax) given as 3.8 mm(3.8 x 10-3 m)peak displacement volume VD SD Xmax 1.24 x 10-4 m3(124 cm3)power rating given as 70W RMS

Sealed Box Design Example constraints – driver resonance frequency (FS) must always be lower than thatof the system (FC)– α must be at least 3– QTS must be lower than highest acceptable QTC– VAS must be at least several times larger than the enclosuresize (volume)– select most desirable combination of FC and QTC that satisfiesFC / QTC FS / QTS and then calculate α VAS / VBcalculations– QTS (0.77)(1.89)/(0.77 1.89) 0.547– based on α 3 requirement, FC/FS 2 FC 62– FS / QTS 56.7 FC / QTC for FC 62 get QTC 1.1(“Chebychev Alignment” maximum efficiency)– internal volume of box needed based on αα VAS / VB (want α 3) VB 22 liters (0.022 m3)– yields approximate size of: 0.28 m x 0.28 m x 0.28 m

Vented Box Design Example alignments (choice depends on Qts)– QB3 – quasi 3rd order Butterworth– SBB4 – 4th order Butterworth (“maximally flat”)– C4 – 4th order Chebychev need to determine port size/length

- unitless measurement, characterizing the mechanical damping of the driver, i.e., losses in the suspension (surround and spider) These values can be determined by measuring the input impedance of the driver, near the resonance frequency, at small input levels for which the mechanical behavior of the driver is effectively linear.

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