Sage Model Notes LoudspeakerVoicecoil.stl

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1 Sge Model Notes LoudspekerVoicecoil.stl D. Gedeon 24 June 2012 A model of voice-coil type liner ctutor commonly used for driving loudspekers. A circulr coil moves xilly bck nd forth within rdil mgnetic field between inner nd outer pole pieces. The coil moves prllel to the pole fces in the direction shown by the red rrow. The mgnets produce mgnetic field in the direction of the green rrows which turns rdilly in the gp between pole pieces: Mgnet Ring Outer Pole Piece Inner Pole Piece Moving Coil Bck Iron Flux Pth The voice coil is longer thn the ir gp so tht it remins entirely within the gp during norml opertion nd produces liner vrition of ctutor force with electricl current, independent of coil position. The speker cone ttched to the coil nd outer structure tht holds everything in lignment is not shown. 1

2 The Sge root model looks like this: A current source (top row) drives electricl current through the coil within the voice-coil ctutor submodel. A constrined piston fixed iron reference nchors the iron nd mgnet ssembly nd the moving coil drives free-piston speker cone which is lso supported by suspension spring so tht the coil stys centered in the gp nd suspension dmper to limit the mplitude t resonnce, when the coustic loding is low. The speker cone imposes n dditionl dmping force bsed on the coustic loding. More on tht lter. 2

3 Within the trnsverse-coil ctutor submodel re these components representing the effective outer mgnetic circuit consisting of the permnent mgnet nd iron: Inside the left-turned mgnetic gp re moving EM continer nd pole pir components: The voice-coil resides within the moving EM continer: The terminology left-turned nd left-turning refers to the direction the inner pole piece directs the incoming mgnetic flux from the outer pole piece. Which is the direction shown by the rrows in the bove icon or in the negtive x direction. This mtters becuse the mgnetic flux linked through the coil is the x-directed flux in the inner pole piece, which increses towrd the negtive x direction. This leds to symmetries in the coil force with displcement giving the force DC bis, even though the coil current my hve no DC component. There re lso right-turned nd right-turning mgnetic gp nd voice coil components vilble in Sge s component plette. 3

4 There re user-defined inputs defined in the voice-coil ctutor submodel bsed on the symbols in the dimensioned picture below. Hm G Rp Rm Lc t i Lm t i Inputs Lmg pole & mgnet length (m) 2.000E-02 Rmg mgnet inner rdius (m) 4.000E-02 Rpole inner pole rdius (m) 2.500E-02 Gp ir gp (m) 4.000E-03 Hmg mgnet rdil height (m) 2.000E-02 thkiron iron thickness (m) 1.000E-02 Lcoil coil outer length (m) 2.000E-02 User defined vribles clculte the effective iron pth re (cross section t the inner mgnet rdius), totl iron pth length nd mgnet cross section re s follows: Outputs Airon iron pth re 1.257E-03 Pi * Rmg * thkiron Liron combined iron pth length 5.850E-02 (Rmg - 0.5*Rpole) + Hmg + (Rmg - (Rpole + Gp)) Amg mgnet cross section re 6.283E-03 Pi * (Sqr(Rmg + Hmg) - Sqr(Rmg)) The iron pth re is tken s the iron circumferentil section re t the inner pole rdius Rp. The iron pth length is the sum of three rdil segments: in the bck plte from Rp/2 to Rm, on the front plte from Rp + G to Rm, plus two segments from Rm to mgnet centroid of combined length Hm. The mgnet cross section re is the true nnulr re. In terms of the bove, the inputs for number of components re recst: 4

5 Iron Pth Recsts Lpth = Liron Apth = Airon Permnent Mgnet Recsts Lpth = Lmg Apth = Amg Left-turned Mgnetic Gp Recsts Zgp = Gp Wpole = 2*Pi * (Rpole + 0.5*Gp) Lpole1 = thkiron Moving EM Continer Recsts Length = Lcoil Offset = -0.5*(Lcoil - thkiron) The offset defines the coil position within the ir gp when the reltive displcement of the moving EM continer (output Xrel) is zero. The bove vlue centers the coil in the gp t zero position. In the left-turning voice coil the input tht estblishes the clernce between the wire nd pole fces is ZthkRel thickness frction of prent Zgp [0, 1] 7.500E-01 Acousticl Loding The speker cone component connected to the voice coil (vi force connection) is reciprocting piston component with these inputs: Inputs Mss reciprocting mss (kg) 1.500E-01 Dcone cone dimeter (m) 2.500E-01 The Mss represents the mss of the moving cone + coil, without the effective mss produced by the coustic loding. Dcone is user-defined input representing the effective dimeter of the moving cone (dimeter tht displces the sme volume s the ctul cone for given displcement). The forcing function input FF provides wy to include n externl force in the reciprocting mss eqution of motion (Newton s lw F = M A). In this model tht force is set to represent the cousticl loding on the cone. Morse 1 provides n pproximte expression for the cousticl impednce for the cse of sound rdition from piston in wll (speker enclosure). The impednce for circulr piston of rdius r is written in the form p c R ix u p nd u re the pressure nd velocity phsors (complex mplitudes), ρc is the men ir density times the speed of sound. R nd X re the resistive nd inertil components of complex fctor tht pproch 1 nd 0 respectively (impednce of trveling plne wve) for speker cone rdius tht is lrge compred to the wvelength. Otherwise Morse 1 In, D.H. Menzel, Fundmentl Formuls of Physics, Volume 1, Dover, (1960), p

6 provides the following pproximtions in terms of the wve number k=ω/c (wvelength λ=2π/k). R kr, kr 1 1, kr 1 8kr X, 3 kr 1 2, kr kr 1 In terms of externl force mplitude F nd cone re A the cousticl impednce my be written F ca R ix Where u is the velocity phsor of the moving cone. Expnded in terms of velocity rel nd imginry prts u r nd u i the force mplitude is So input FF is recst to: FF = 0.000E (-FFmp) Amp (FFrg) Arg F ca u R u X u i R u X u r In other words the mplitude is recst to -FFmp nd the phse to FFrg which re defined t the end of the following set of user-defined vribles: Acone cone re 4.909E *Pi * Sqr(Dcone) Csound speed of sound 3.500E RhoCir Rho * C for ir 4.025E * Csound kr wve number * cone rdius 2.244E-01 2*Pi*Freq/Csound * 0.5*Dcone R resistive impednce term 2.518E-02 Min(0.5*Sqr(kr), 1) X inertil impednce term 1.905E-01 Min(8*kr/(3*Pi), 2/(Pi*kr)) Ur rel prt velocity E+00 2*Pi*Freq * FX.Sin.1 Ui imginry prt velocity E-01 2*Pi*Freq * FX.Cos.1 FFr rel prt coustic force E+00 RhoCir*Acone*(R*Ur + X*Ui) FFi imginry prt coustic force 6.040E+00 RhoCir*Acone*(R*Ui - X*Ur) FFmp coustic force mplitude 7.363E+00 Sqrt(Sqr(FFr) + Sqr(FFi)) i i r 6

7 FFrg coustic force phse 1.249E /Pi * Arg(FFr, FFi) Wcoust cousticl output power 9.358E-01 W.Men How does the rdited cousticl power depend on frequency? According to the resistive impednce component R bove, the cousticl loding imposes dissiptive force on the cone tht in the limit of high frequency (plne wve impednce) is proportionl to nd in phse with the cone velocity, like dmper. At moderte frequencies (kr < <1) the dissiptive force is proportionl to ω 2. But the rdited power is proportionl to the product of force nd velocity mplitudes nd the velocity mplitude is minly determined by the inerti of the moving mss. This mens tht cone velocity mplitude tends to decrese in proportion to 1/ ω 2 for constnt current mplitude (nerly constnt coil force mplitude). The net result is tht t moderte frequencies the rdited cousticl power (product of dissiptive force nd velocity mplitudes) is nerly independent of frequency, the holy gril of loudspeker designers. At high frequencies the resistive impednce component R stops incresing nd the rdited power drops off s 1/ ω 2. This behvior cn be seen in the plots below. A different spin on the result of chieving frequency independent cousticl power is tht it presents loding on the coil tht gurntees low electricl efficiency. This cn lso be seen in the plots below. Energy Blnce It is helpful to consider the energy blnce in the sttionry prts seprte from the moving coil. The following tble ccounts for the moving coil energy blnce within the ir gp for the model running t 100 Hz with current mplitude 5.0 mp. Input power from current source (Fwe) Coil I 2 R loss (Wdissip) Mechnicl power output to suspension dmper (W) Acousticl power output to speker cone (W) Net power leving ir gp Power W E E E E E+01 This sme power should be equl to the men vlue for the FWm (mgnetic power inflow) output for the pole pir component, which is E+01. The difference of bout 0.2 W my be round-off error or smll numericl energy lek. The next tble ccounts for the energy dissiption of tht power in the sttionry mgnetic components. Power W Iron pth eddy-current loss (Weddy) 2.295E+01 Iron pth hysteresis loss (Whyst) 1.149E+00 permnent mgnet eddy-current loss (Weddy) 1.336E-04 Totl losses 2.41E01 So energy is conserved within bout 0.2 W. The eddy current losses depend on the lmintion thicknesses for the iron nd mgnet components. For loudspeker ctutors the time vrition of mgnetic flux in the iron nd mgnets is reltively low so there re no lmintions per sy. So the lmintion thickness 7

8 inputs for the iron pth is recst to t i nd the permnent mgnet to W m (in the bove drwing) which re the minimum dimensions of the components when looking t typicl cross section norml to the mgnetic flux direction. The thklm inputs re defined in the object components inside the iron pth nd permnent mgnet components. Electricl Impednce Mp The model is set up to mp frequency (input Freq) over the rnge 20 to 2,000 Hz. The nturl frequency of the speker cone + suspension spring is 41 Hz ccording to the formul 2 f K M The cone mss is M = 0.15 kg nd spring stiffness is K = 1.0E4 N/m. Of interest to loudspeker designers is the electricl impednce t the coil terminls defined s V Z I Where V nd I re voltge nd current phsors. Electricl impednce is combintion of coil resistnce nd inductnce complicted by the reflection of the cone dynmic response into the coil voltge nd current. The mplitude of Z is V 1 /I 1 where V 1 nd I 1 re the voltge nd current mplitudes nd the phse is the difference between voltge nd current phses. The model clcultes electricl impednce vi these user-defined vribles of the current source component: Zmp electricl impednce mplitude 1.435E+01 FDeltV.Amp.1 / FI.Amp.1 Zrg electricl impednce phse E+01 FDeltV.Arg.1 - FI.Arg.1 When mpped over frequency rnge the model produces this electricl impednce curve: 8

9 According to Wikipedi 2 the nominl impednce of loudspeker is 1.15 time higher thn the minimum impednce long the frequency curve. So the loudspeker in the model hs nominl impednce of bout 10.4 ohms (minimum impednce 9.0 ohms). To produce the impednce mp it ws helpful to temporrily reduce the current mplitude to 0.5 mp

10 Rdited Acousticl Power The plot below shows the speker cone cousticl power output for constnt current mplitude of 0.5 mp. A shrp pek t the 41 Hz mechnicl resonnt frequency is eliminted by choice of the dmping coefficient (40 N s/m) of the suspension dmper component. Tht is bout hlf the criticl dmping rtio of 77 Ns/m, given by the stndrd formul D c 2 MK The level re between 80 Hz nd bout 800 Hz is the desirble loudspeker behvior where the resistive impednce component R is growing s the squre of frequency, s discussed bove. The high frequency drop-off is where R = 1. The trnsition is more brupt thn would ctully be the cse becuse of the two-prt pproximtion used to clculte R (speker cone user vrible R bove) 10

11 Efficiency The loudspeker electricl efficiency cn be seen in this plot: Electricl efficiency is defined s time-verge electricl input power / cousticl power output, which is vilble in the model vi user-defined vribles Wcous in the speker cone, Welec in the current source nd Effic in the root model. The pek efficiency is bout 0.75% t bout 400 Hz. The efficiency is necessrily low becuse of the behvior of the cousticl loding. If the voice coil were loded by simple dmper nd driven t its resonnt frequency the efficiency would be much higher. Looking t it nother wy, the efficiency is low becuse the electromgnetic dissiptions (minly the coil resistnce dissiption) re reltively constnt but the rdited power is very low. Solution Grid You cn understnd more bout the model by dumping the solution grid for the two-pole mgnetic gp component, which contins the grids for the moving EM continer, the voice coil, the coil object nd the pole pir. For exmple, the plot below shows the liner mgnetic flux distribution in the coil-reltive reference frme t vrious times s it moves through the mgnetic gp. 11

12 The mgnetic flux is highest for the prt of the coil between the pole fces nd drops off on either side ccording to the fringing flux formultion in Sge. The reson the loction of the highest mgnetic flux moves bck nd forth is becuse the plot is in the coil reference frme from which the outer mgnetic structure ppers to be moving bck nd forth. The effect of the coil electricl current shows up in the sloping mgnetic flux for the prt of the coil between the pole fces. When the current is zero (drk blue curve) the mgnetic flux is produced solely by the permnent mgnet nd is uniform long the pole fces. When current is not zero it produces mgnetic potentil difference tht increses in mgnitude from zero t the left (negtive) end of the coil to mximum t the right end nd beyond. This drives sloped mgnetic flux long the pole fces, either upwrd or downwrd depending on the direction of the current (mgnetic potentil difference ). 12

Version 001 HW#6 - Electromagnetism arts (00224) 1

Version 001 HW#6 - Electromagnetism arts (00224) 1 Version 001 HW#6 - Electromgnetism rts (00224) 1 This print-out should hve 11 questions. Multiple-choice questions my continue on the next column or pge find ll choices efore nswering. rightest Light ul