Lumped Parameter Analysis of a Dynamic Loudspeaker

1 Lump edParameterAnalysisofaDynamicLoudsp eakerPaavoJumppanenApril8,2013 Thelowfrequencyb ehaviourofdynamicloudsp eakerdriversiswellundersto o ,particularlyventedloudsp Loudsp eakersinVentedBoxes pap ceedingsandhavetoadmitthatI ndithardtofollow,principallyb ecauseitisbuiltaroundthecontructionofequ ivalentcircuitstorepresentdriverb delling(itisn'tcoveredinthepap er)makesitdi culttounderstandwheretherevariousimp erent,andIb elieve,morereadilyundersto o delsofdynamicloudsp eakersgenerallyassumethatthedriverop eratesinwhatisreferredtoasthepistonop ehavesasarigidb o o diesanditisessentiallyimp os-sibletomakeap erfectpistonbutforlowfrequenciestherigid b o ointatwhichconebreakupb ecomesadominantfactorindriverresp onsegenerallyo ,forthee ectoftheb oxontheresp onseitisassumedthatthewavelengthofsoundi slargecomparedwiththephysicaldimensionso ftheenclosure.

2 Whichisgenerallytruefortherangeoffrequen ciesforwhichtheb oxin uencesthedriverresp (themassofairincontactwiththecone)andthe radiationresistance(energytransmit-tedto airassoundwaves)islump edinwiththemo etweensp eedofpropagationofsoundinairVair,wavelen gth andwavefrequencyf,namelyVair=f 1 DKMFxFigure1:mechanicalmo delofloudsp eakerTheMo delWestartwiththerigidb o dyassumptionandconsiderthemechanicalcomp o-nentsresp onsibleforthedynamicresp ersystem,theconeandthevoicesupplyingthem ovingmass,thesusp ension(includingthespiderandringsurround )providingthesti edparametermechanicalmo delin ndthat,F=Kx+Md2xdt2+Ddxdt(1)whereFisthef orceappliedtothevoicecoil,xisthedisplace mentofthecone,Kisthesusp ensionsti ness, netheLaplaceTransformofx(t)withzerointia lconditionsas,X(S) =L(x(t))(2)Itcanb eshownthat,L(dx(t))

3 Dt)=SX(S)(3)ormoregenerally,L(dnx(t)dtn) =SnX(S)(4)Thisidentitycanthenb eusedtotransformourdi erentialequationinthetimedomainintoap olynomialinS,thecomplexfrequencydomain,F (S) =(MS2+DS+K)X(S)(5)2V(S)Eb(S)I(S)RLeFigur e2:ExcitationCircuitThisshorthandwayofde alingwithdi eakerwecanrepresenttheexcitationcircuitw iththestructureshownin edanceofaninductor3isSLitfollowsthat,I(S ) =V(S) Eb(S)R+SLe(6)whereVisthevoltageappliedto thevoicecoil,Risthevoicecoilresistance, ortionaltocurrentgivingF(S) =KfI(S) =KfV(S)R+SLe KfEb(S)R+SLe(7)whereKfistheconstantofpro p ortionalityb ortionaltoconevelo citygivingEb(S) =KgSX(S)(8)whereKgistheconstantofproptio nalityb etweenbackEMFandconevelo nd,X(S)V(S)=KfSKgKf+F(S)X(S)(R+SLe)(9)

4 Wealsonotethatacousticsoundpressureispro p ortionaltotheaccelerationoftheconegiving Aspl(S) =KaS2(10)2 Thevoltage,v,requiredtocreateacurrent owofithrougharesistorofresistanceRohmsis givenbyv=iR3thecompleximp edanceofaninductorfollowsfromthedi erentialequationgoverningitsop eration,namelyv= (S) =SL3V(S) + 1R+SLeExcitationcurrentKfForce1MS2+DS+KD isplacementS2 AccelerationKaAspl(S)KaKgBackEMFSVelo cityFigure3:DynamicLoudsp eakerBlo ckDiagramwhereKaistheconstantofprop ortionalityb ckdiagramof okingatthevoicecoilcomp onentinmoredetail,letusassumethemagnetic induction/ uxdensityinthep oleairgapisB, nd,F=ilB=in DB(11)fromwhichwecandeducethat,Kf=n DB(12)FromFaradayslawwehave,Eb=nd dt(13)where isthe uxwithachangeofdisplacementis,d = DBdx(14)giving,Eb=n DBdxdt(15)Fromwhichitisclearthat,Kg=n DB=Kf(16)4 Loudsp eakerDisplacementResp onseFrom gure3weseethatthemo delisanegativefeedbacklo onse,X(S)V(S)=Kf(R+SLe)

5 (MS2+DS+K) +Kf2S(17)Tosimplifytreatmentofthedisplac ementrep onseofthedriverweshallignorethee (S)V(S)=KfMRS2+(DR+Kf2)S+KR(18)Wecanre-w ritethisresp onseinthenormalised,X(S)V(S)=(1Kf sQe) s2S2+( sQm+ sQe)S+ s2(19)where, sQm=DM sQe=K2fRM s2=KMWeshallellab orateonthesetermslaterwhendiscussingdriv erimp onseisasecondorderlowpassresp onseisobtainedbysubstitutingS=j giving4,X( )V( )=(1Kf sQe) s2j ( sQm+ sQe)+ s2 2(20)Loudsp eakerSoundPressureResp onseTheradiatedsoundpressureisprop gure3andequation19weseethat,Aspl(S)V(S)= (Ka sKfQe)S2S2+( sQm+ sQe)S+ s2(21)Weseethesoundpressureresp onseisasecondorderhighpassresp onseisobtainedbysubstitutingS=j giving,Aspl( )V( )=(Ka sKfQe) 2j ( sQm+ sQe)+ s2 2(22)4jistheimaginarynumb erde nedbytheidentityj= ,MathematiciansandmechanicalEngineersuse itoindicatetheimaginarynumb.

6 Imp edancePresentedbyaLoudsp eakerMountedinanIn niteBa eLoudsp eakerImp edanceFromOhmslaw,Z(S) =V(S)I(S)(23)I(S) =V(S) Eb(S)R+SLe(24)Therefore,Z(S) =V(S)V(S) Eb(S)(R+SLe) =11 Eb(S)V(S)(R+SLe)(25)Substitutingequation 8we nd,Z(S) =11 KgSX(S)V(S)(R+SLe)(26)Substitutingthedis placementresp onseofequation17intotheab ovewe nd,Z(S) =(R+SLe)(MS2+DS+K)+Kf2S(R+SLe) (MS2+DS+K)(R+SLe)(27)Z(S) =S3Le+S2(R+LeDM)+S(RDM+LeKM+Kf2M)+RKMS2+ DMS+KM(28)Toseehowthistranslatesintoaneq uivalentelectricalcircuit,considerthenet workin edanceofthisnetwork5is,5thecompleximp edanceofacapacitorfollowsfromthedi erentialequationgoverningitsop eration,namelyi= (S) =1SC6Z(S) =R+11SL+1r+SC+SLe(29)Z(S) =S3Le+S2(R+LeCr)+S(1C+RCr+LeLC)+RLCS2+S1 Cr+1LC(30)Comparingequation28andequation 30wecanseethatifwecho ose,1Cr=DM(31)1LC=KM(32) ndthatweneed,(RDM+LeKM+Kf2M)=(1C+RCr+LeL C)(33)Fromwhichitfollowsthat,Kf2M=1C(34)

7 Thenormalisedformofasecondorderp olynomialinSis,S2+S oQ+ o2 Wecanpro duceanormalisedformfortheimp edanceifweintro ducetheterms sastheresonantfrequency6ofthedriver,Qmas themechanicalqualityfactorforthedriver,Q eastheelectricalqualityfactorand easthep ne, sQm=1Cr=DM(35) sQe=K2fRM(36) s2=1LC=KM(37) e=RLe(38)Normalisingourimp edancefunctionwe nd,6 Beawarethat sistheangularresonantfrequencywithunitso fradiansp s= 2 (S) =RS3(1 e)+S2(1 + sQm e)+S( s2 e+ sQm+ sQe)+ s2S2+S( sQm)+ s2(39)Fromequations34,36and35wecanshowth at,r=RQmQe(40)Re-arrangingequation36give s,C=Qe sR(41)Finally,substitutingCab oveintoequation37andre-arranginggives,L= R sQe(42)IfweIgnorethee ectofLe,whichistypicallysmallatdriverres onance,wecansimplifytheimedanceto,Z(S) =R S2+S( sQe+ sQm)+ s2S2+S( sQm)+ s2 (43)Fromwhichweseethatattheresonantfrequ encythemagnitudeofthedriverimp edanceisatamaximumof,|Z(j s)|=Zmax=RQm+QeQe(44)

8 Thisrelationshipcomesinhandyfordetermini ngtheb oxQofanenclosure, ortantpartofpassivecrossoverdesignisthec onstructionofanetworktoequalisetheimp edancewillresultinacombined lterresp onsethatdeviatesfromthedesiredresp edanceofthisnetworkis,Zin=11R+Zl+1R+Zc(4 5)Zin=R2+ZlZc+R(Zl+Zc)2R+Zl+Zc(46)8 RZcRZlZinFigure5:GeneralisedConjugateImp edanceRcCeLcCcFigure6:Imp edanceEqualiserNetworkZcwhereZlisthecomp leximp edancewewishtoequaliseandZcisthecomp en-sationimp eRfromwhichitfollowsthatforthistob etruethen,Zc=R2Zl=R2Yl(47) eakerimp edance,considerthenetworkin edanceofthisnetworkis,Zc=11 SCe+1Rc+SLc+1 SCc(48)Zc=Rc+SLc+1 SCcSCe(Rc+SLc+1 SCc)+ 1(49)Fromequations30and47wecanseethat,R2 Yl=R2SL+R2r+SCR2 SLe(1SL+1r+SC)+ 1(50)9 Comparingthetoplinesofequation49withequa tion50weseethat,Rc=R2r(51)Lc=CR2(52)Cc=L R2(53)

9 Comparingtheb ottomlinesofequation49withequation50wese ethat,CeCc=LeL(54)CeRc=Ler(55)CeLc=LeC(5 6)Thisallholdstrueif,Ce=LeR2(57)(58)Summ arisingtheresultswehave,10 ModelimpedancecomponentvaluesConjugatelo admatchcomponentvaluesRLeLrCRRcCeLcCcr=R QmQeRc=R2r=RQeQmC=Qe sRCc=LR2=1 sQeRL=R sQeLc=CR2=QeR sCe=LeR2 TheResp onseofaDriverinaSealedBoxMountingadriver inasealedb oxaddstwoextracomp onentstothemechanicalmo delof gure1:anextradampingterm,Db,andanextrast i nessterm, ondingdrivertermssoinessence,asealedb oxsimplyincreasesthedriversusp ensionsti ,DM DM+DbM= sQm+ bQb(59)andKM K+KbM= s2(1 +VASVb)= b2(60)ThenormalisedformoftheKbtermstemsf romthede nitionofVAS,thevolumeequivalentsti nition,11Kb=KVASVb(61)Equation60showstha tb oxraisestheresonantfrequencyofthedriverw iththenewresonanceb ecoming.

10 B= s 1 +VASVb(62)orputinanotherway,theb oxvolumerequiredtoraisetheresonanceto bis,Vb=VAS( b s)2 1(63)Thenewdisplacementandso