Loudspeaker Voice-Coil Inductance Losses: Circuit Models, Parameter Estimation, and Effect on Frequency Response

Transcription

1 44 JOURAL OF THE AUDIO EGIEERIG SOCIETY, VOL. 50, O. 6, 00 JUE Loudspeaker Voce-Col Inductance Losses: Crcut Models, Parameter Estmaton, and Effect on Frequency Response W. Marshall Leach, Jr., Professor Georga Insttute of Technology School of Electrcal and Computer Engneerng Atlanta, Georga USA Abstract When the seres resstance s separated and treated as a separate element, t s shown that losses n an nductor requre the rato of the flux to mmf n the core to be frequency dependent. For small-sgnal operaton, ths dependence leads to a crcut model composed of a lossless nductor and a resstor n parallel, both of whch are frequency dependent. Mathematcal expressons for these elements are derved under the assumpton that the rato of core flux to mmf vares as ω n 1,wheren s a constant. A lnear regresson technque s descrbed for extractng the model parameters from measured data. Expermental data are presented to justfy the model for the lossy nductance of a loudspeaker voce col. A SPICE example s presented to llustrate the effects of voce-col nductor losses on the frequency response of a typcal drver. I. ITRODUCTIO For small-sgnal operaton, the voce col of an electrodynamc loudspeaker drver can be modeled by three elements n seres a resstance, a lossy nductance, and a dependent voltage source representng the back emf generated when the daphragm moves [1]. The crcut s shown n Fg. 1, where R E s the resstance and L E s the nductance. The back emf s gven by B u D,whereB s the magnetc flux n the ar gap, s the effectve length of wre that cuts the flux, and u D s the mechancal velocty of the daphragm. The back emf due to the daphragm moton exhbts a band-pass effect that decreases toward zero as frequency s ncreased above the fundamental resonance frequency of the drver. At the hgher frequences, the mpedance s domnated by the nductance. Fg. 1. Equvalent voce-col crcut. When the seres resstance of the voce col s separated and treated as a separate element, the lossy nductance can be modeled at any frequency by a crcut consstng of a lossless nductor n parallel wth a resstor []. If the frequency s changed, the values of both the nductor and the resstor change. In [3], t s shown that eddy current losses n the magnet structure cause the mpedance of the lossy nductor to be of the form Z = K jω. In [4], expermental data s presented whch shows that ths model fals to predct the hgh-frequency mpedance of many drvers. An emprcal model s descrbed for whch the mpedance of the lossy nductor s assumed to be of the form Z = K r ω Xr + jk ω X. An expermental method for determnng the model parameters s descrbed that s based on mpedance measurements at two frequences. The lossy nductance model of [3] requres one parameter. The model of [4] requres four. In the followng, a model s derved whch requres two parameters. A lnear regresson method for determnng these from measured voce-col mpedance data s developed and an example s presented. A SPICE model for the lossy nductor s descrbed and a SPICE smulaton s used to llustrate the effect of the nductor losses on the frequency response of a drver. II. IDUCTOR FUDAMETALS The analyss presented here assumes an nductor that s wound wth wre that exhbts zero resstance. When the analyss s appled to the lossy nductance of a loudspeaker voce col, t s assumed that the seres resstance of the voce col has been separated and s treated as a separate element. Although both the large-sgnal and small-sgnal behavors of nductors are revewed n ths secton, the model developed for the lossy nductor s strctly vald only for small-sgnal operaton. Fgure llustrates an nductor consstng of turns of wre wound on a rectangular core. The total flux λ lnkng the col s gven by λ = ϕ,where s the number of turns of wre and ϕ s the flux lnkng a sngle turn. The voltage across the col s gven by v = dλ dt = dϕ (1) dt The magnetc propertes of the core materal determne the relatonshp between the current n the col and the mpressed voltage. These propertes are usually descrbed by a plot of the flux ϕ versus the mpressed magnetomotve force or mmf

2 LOUDSPEAKER VOICE-COIL IDUCTACE LOSSES s for the lnear core. The flux lags the mpressed voltage by 90 o. The current lags the voltage by less than 90 o, ncreasng to 90 o for a lossless core. The current s for the nonlnear core. The current waveform s no longer snusodal. It exhbts a peak value that s hgher by a factor of. Anncreasenthe appled voltage causes ths peak to ncrease rapdly, causng the nductor to approach a short crcut as the core saturates. Fg.. Inductor consstng of turns of zero-resstance wre wound on a magnetc core. gven by F =,where s the current n the col. Two such plots are shown n Fg. 3(a), where t s assumed that the mmf vares snusodally wth tme and has a mean value of zero. The arrows ndcate the drecton of moton around the curves as tme ncreases. Curve 1 assumes the core materal s lnear. In ths case, the curve s an ellpse. For a lossless core, the ellpse degenerates nto a straght lne. All magnetc materals exhbt a nonlnearty that causes the flux to exhbt a saturaton effect as the mpressed mmf s ncreased above some value. Such a plot s shown n curve, where the flux s assumed to saturate at 1/ of the peak value n curve 1. The curves are often referred to as hysteress loops. Fg. 4. Plots of voltage v, flux ϕ, current 1 for the lnear core, and current for the nonlnear core versus tme. In the followng, a lnear magnetc materal s assumed. Otherwse, the defnton of nductance would be mpossble and phasor analyses would be precluded. For a lnear core, the phasor form of Eq. (1) s V = jωϕ = jπfϕ () Fg. 3. (a) Plots of flux versus mmf. (b) Plots of mmf versus flux. The curves n Fg. 3(a) assume that the mmf s the ndependent varable. Because F =, t follows that the curves are plotted for a current source exctaton. If a snusodal voltage s mpressed across the col, t follows from Eq. (1) that the flux lnkng each turn s determned by the voltage. Thus the flux s the ndependent varable for a voltage source exctaton. In ths case, the plots of F versus ϕ are shown n Fg. 3(b). Curve 1 for a lnear core materal remans an ellpse. However, curve for a core materal that exhbts flux saturaton effects shows a rapdly ncreasng mmf as the flux s ncreased. For the assumed flux saturaton factor of 1/, the peak value of the mmf s twce the peak value n curve 1. Fgure 4 llustrates the voltage, flux, and current waveforms for a snusodal voltage mpressed across the col. The current where ω =πf s the radan frequency. For V a constant, t can be seen that ϕ s nversely proportonal to the product f. It follows that any effects of core saturaton are reduced f the frequency of the mpressed voltage s ncreased or f the number of turns of wre s ncreased. Ths result plays an mportant role n transformer desgn f core saturaton problems are to be avoded. The lower the frequency of operaton, the more turns of wre must be used, causng low-frequency transformers to be larger than hgh-frequency counterparts. It also explans why a step-down transformer cannot be used n reverse,.e. wth the prmary and secondary reversed. For a lnear magnetc materal, a snusodal voltage mpressed across the col results n snusodal current, flux, and mmf. Let the flux and mmf be gven by ϕ (t) =ϕ 1 cos ωt =Re[ϕ 1 exp (jωt)] (3) F (t) =F 1 cos (ωt + θ) =Re[F 1 exp (jθ)exp(jωt)] (4) where θ s the angle by whch the mmf leads the flux, or alternately the angle by whch the flux lags the mmf. In general, θ s a functon of the frequency ω. The correspondng voltage and current are gven by dϕ (t) v (t) = dt = ωϕ 1 sn ωt = Re[jωϕ 1 exp (jωt)] (5)

3 444 JOURAL OF THE AUDIO EGIEERIG SOCIETY, VOL. 50, O. 6, 00 JUE (t) = 1 F (t) cos (ωt + θ) = Re exp (jθ)exp(jωt) = F 1 F1 It follows that the phasor voltage and current, respectvely, are gven by V = jωϕ 1 (7) I = F 1 exp (jθ) =F 1 (cos θ + j sn θ) (8) For a constant mpressed voltage, the current n the col s proportonal to the admttance Y (jω) gven by Y (jω) = I V F 1 = exp (jθ) jω ϕ µ 1 F 1 sn θ = ϕ 1 ω j cos θ (9) ω It follows that the equvalent crcut of the nductor can be represented as a parallel resstor and nductor gven by R = ω ϕ 1 F 1 sn θ (6) (10) L = ϕ 1 (11) F 1 cos θ Because R and L must be postve, t can be seen that the angle θ must satsfy the condton 0 θ 90 o. These equatons are used n the followng as the bass for the crcut model for the lossy nductance of a loudspeaker voce col. The key n the development of the model s the frequency dependence of the rato of flux to mmf n the core,.e. n the frequency dependence of ϕ 1 /F 1. III. THE LOSSY IDUCTOR MODEL The author teaches a senor electve audo engneerng course at Georga Tech where students are requred to brng a loudspeaker drver nto the laboratory and measure ts small-sgnal parameters. After the acquston of equpment that provded the capablty of makng detaled automated measurements of voce-col mpedance at hgh frequences, t was notced that the phase of the mpedance of drvers approached a constant at hgh frequences after the seres resstance of the col and the motonal mpedance term are subtracted. For a lossless vocecol nductance, ths phase should be 90 o. However, expermentally observed values were usually n the 60 o to 70 o range. By concdence, t was observed that the phase could be predcted from the slope of the log-log plot of the magntude of the mpedance versus frequency. Ths slope, when multpled by 90 o, predcted the phase. Ths nterestng and puzzlng relatonshp was explaned when the author remembered Bode s gan-phase ntegral [5] from an undergraduate course n control systems. Ths ngenous ntegral s the bass of the lossy nductor model that s descrbed n ths secton. By Eq. (9), the admttance of the lossy nductor has a magntude and phase gven by Y (jω) = p Y (jω) Y (jω) = F 1 ω ϕ 1 (1) β = arg[y (jω)] ½ ¾ Im [Y (jω)] = tan 1 Re [Y (jω)] = θ π (13) If the rato of ϕ 1 to F 1 s ndependent of frequency, t can be seen that Y (jω) s nversely proportonal to ω. In ths case, the slope of the plot of ln Y (jω) versus ln (ω) s 1. Because there s no evdence that a two-termnal passve lumped element network can have an admttance or mpedance transfer functon that s not mnmum phase, t wll be assumed here that Y (jω) represents a mnmum-phase transfer functon. In ths case, the phase β s related to the magntude by Bode s gan-phase ntegral β = 1 Z dα ³ π du ln coth u du (14) where α =ln Y (jω) and u =ln(ω). For the case ϕ 1 /F 1 = aconstant, Y (jω) 1/ω n Eq. (1), and t follows that dα/du = 1. In ths case, Bode s ntegral predcts β = π/, where the relaton Z ³ coth u du = π (15) ln has been used. By Eq. (13), β = π/ results n θ =0. Thus by Eqs. (10) and (11), R s an open crcut and L s a constant. In ths case, the nductor s lossless. It follows that losses requre the rato of flux to mmf to be frequency dependent. For a lossy nductor, the most general model for the frequency dependence of the rato of flux to mmf s a power seres n ω. To model an mpedance whch has a phase that s ndependent of frequency, t follows from Bode s ntegral that the power seres must have only one term. It s assumed here that ϕ 1 /F 1 = Kω n 1 /,wheren and K are constants. It has been observed that ths choce leads to excellent agreement wth expermental data and an example s presented n the followng whch llustrates ths. In ths case, t follows from Eq. (1) that Y (jω) =1/ (Kω n ) so that dα/du = n and Bode s ntegral predcts β = nπ/. By Eq. (13), ths results n θ =(1 n) π/.thus,eq.(9)fory (jω) can be wrtten Y (jω) = = = 1 exp jωkωn 1 1 (jω) n K 1 h ³ nπ cos Kω n j (1 n) π j sn ³ nπ (16) It follows that the equvalent crcut of the nductor conssts of a parallel resstor R p and nductor L p gven by R p = Kωn cos (nπ/) (17)

4 LOUDSPEAKER VOICE-COIL IDUCTACE LOSSES 445 L p = Kωn 1 (18) sn (nπ/) For both R p and L p to be postve, n must satsfy 0 n 1. For n =1, R p = and L p s ndependent of ω. Forn =0, L p = and R p s ndependent of ω. It can be concluded that the losses ncrease as n decreases, causng the nductor to change from a lossless nductor nto a resstor as n decreases from 1 to 0. The equvalent crcut of the nductor s shown n Fg. 5(a). the range of frequences for the data was much lower than the range of nterest n loudspeaker drvers, ths value of n s close to values observed by the author for some drvers. In partcular, t corresponds exactly to the value measured for one sample of an 18 nch JBL model 41H Professonal Seres drver. Fg. 6. Plot of log (P ) versus log (f ) for expermental transformer core loss data presented n [6]. Fg. 5. Equvalent crcuts of the lossy nductor. (a) Parallel model. (b) Seres model. From Eq. (9), t follows that the mpedance of the lossy nductor s gven by Z (jω) = 1 Y (jω) = K (jω) n ³ nπ ³ nπ = Kω hcos n + j sn (19) Thus the equvalent crcut can also be represented by a seres resstor R s and nductor L s gven by ³ nπ R s = Kω n cos (0) ³ nπ L s = Kω n 1 sn (1) For n =1, R s =0and L s s ndependent of ω. For n =0, L s = 0 and R s s ndependent of ω. The seres equvalent crcut of the nductor s shown n Fg. 5(b). For n =1/, the expresson for Z (jω) reduces to the one derved n [3]. For ths case, the real and magnary parts of Z (jω) are equal and vary as ω. For a snusodal flux havng a peak ampltude ϕ 1, the peak ampltude of the mpressed voltage s ωϕ 1. The average power dsspated n the nductor s thus gven by P = (ωϕ 1) R p = (ϕ 1 ) cos (nπ/) ω n () K As an example, Fg. 6 shows a log-log plot of expermental transformer core loss data gven n [6] for losses n lamnated slcon steel for a constant mpressed flux at four frequences. The straght lne approxmaton to the data has a slope of 1.4. For ths case, t follows that n = 1.4 = Although Although the parallel model corresponds to the tradtonal model for a lossy nductor, the seres and parallel models are equvalent. The parameters K and n for a drver can be obtaned from measured voce-col mpedance or admttance data. The method descrbed n the followng uses mpedance data. IV. IDUCTOR PARAMETER ESTIMATIO In general, the voce-col mpedance of a drver on an nfnte baffle can be wrtten [7] Z VC (s) = R E + Z L (s) R ES (1/Q MS )(s/ω S ) + (s/ω S ) +(1/Q MS )(s/ω S )+1 (3) where s = jω, R E s the voce-col resstance, Z L (s) s the mpedance of the lossy nductor, ω S s the fundamental resonance frequency of the drver, Q MS s ts mechancal qualty factor, and R ES s the amount by whch the mpedance peaks up at resonance. The procedure descrbed n the followng for determnng n and K assumes that R E, R ES, ω S,andQ MS are known. It follows from Eq. (3) that the lossy nductor mpedance Z L (jω) can be wrtten Z L (jω) = Z VC (jω) R E R ES (1/Q MS )(jω/ω S ) (jω/ω S ) +(1/Q MS )(jω/ω S )+1 (4) The natural logarthms of Z L (jω) and the model mpedance Z (jω) gven by Eq. (19), respectvely, are gven by ln [Z L (jω)]=ln Z L (jω) + j arg [Z L (jω)] (5) ln [Z (jω)]=ln(k)+n ln (ω)+j nπ (6) If ln [Z L (jω)] s known over a band of frequences for a partcular drver and the parameters n and K can be determned

5 446 JOURAL OF THE AUDIO EGIEERIG SOCIETY, VOL. 50, O. 6, 00 JUE such that ln [Z (jω)] ln [Z L (jω)]=0over that band, then Z (jω) s an exact model for the nductor mpedance over the band. A method for determnng n and K that mnmzes the mean magntude-squared dfference between the functons s descrbed below. Let Z VC (jω) be measured at a set of frequences and the value of Z L (jω) calculated for each. An error functon can be defned as follows: = X ln [Z (jω )] ln [Z L (jω )] precson when close numbers are subtracted, a major cause s probably the somewhat crude model for Z VC (jω) n Eq. (3) used to calculate Z L (jω). For example, R ES s not a constant, n general. In addton, suspenson creep most lkely affects the ft of the model mpedance around resonance. Also frequency dependence of the suspenson complance and resstance make t dffcult to estmate or even defne a precse resonance frequency. ½ h ln (K)+n ln (ω ) ln Z L (jω ) = X h nπ ¾ + arg [Z L (jω )] (7) For mnmum error between the measured mpedance and the model mpedance, the condtons / n = 0 and / [ln (K)] = 0 must hold. These condtons lead to the solutons n = 1 X ln Z L (jω ) ln (ω ) 1 X ln Z L (jω ) X ln (ω ) + π X arg [Z L (jω )] (8) " ln (K) = 1 X ln Z L (jω ) n X where s gven by ln (ω ) # (9) = X " [ln (ω )] 1 X ³ π ln (ω )# + (30) When these equatons are satsfed, the curves of the magntude and phase of the model mpedance Z (jω) ft those of the measured mpedance Z L (jω) n a mnmum mean-squared error sense. ote that the error s smultaneously mnmzed for a log-log plot of Z L (jω) and a lnear-log plot of the phase of Z L (jω). The above equatons are used n the followng secton to llustrate an applcaton of the lossy nductor model to an example drver. V. A UMERICAL EXAMPLE The drver selected for ths example s the Emnence model Ths s a 10 nch drver havng a 38 ounce magnet and an accordon suspenson. Its measured parameters are R E =5.08 Ω, f S =35. Hz, R ES =3.0 Ω, andq MS =.80. Fg. 7 shows plots of the measured values of Re [Z VC (jω) R E ] and the calculated values of Re [Z L (jω)] defned n Eq. (4). Fg. 8 shows correspondng plots of Im [Z VC (jω) R E ] and Im [Z L (jω)]. In the regon around the resonance frequency, some rpple n the curves for Z L (jω) s evdent. Whle some of ths can be attrbuted to random measurement errors and loss of Fg. 7. Lnear-log plots of Re [Z VC (jπf) R E ] (crcles) and Re [Z L (jπf)] (boxes) versus frequency. Fg. 8. Lnear-log plots of Im [Z VC (jπf) R E ] (crcles) and Im [Z L (jπf)] (boxes) versus frequency. The frequency range chosen for determnaton of n and K from the measured data was the range from khz to 0 khz. Ths regon was chosen to mnmze the effect on the calculatons of a jog n the measured phase below khz. There were 1 measurement ponts n the range. Wth the assstance of Mathcad, the values obtaned from Eqs. (8) through (30) were n =0.688 and K = Fgs. 9 and 10 show the measured magntude and phase of the mpedance Z L (jω) n ths range plotted as crcles and those calculated from Eq. (19) plotted as sold lnes. The magntude approxmaton shows excellent agreement wth the measured values. The measured phase values exhbt a slght rpple about the calculated value of 6 o, some of whch s lkely a remnant of the jog n the phase below khz. The rms phase devaton s 0.66 o, or just over 1%,a fgure whch statstcally ndcates excellent agreement. Fgures 11 and 1 show the magntude and phase of the measured mpedance and the magntude and phase predcted by Eq. (3) for the frequency range from 15 Hz to 0 khz. The measured values are plotted as crcles and the calculated values as sold lnes. The fgures show excellent agreement between the measured values and the values predcted by the model equa-

6 LOUDSPEAKER VOICE-COIL IDUCTACE LOSSES 447 Fg. 11. Lnear-log plots of the measured Z VC (crcles) and the approxmatng functon (sold lne) versus frequency. Fg. 9. Log-log plots of the calculated Z L (crcles) and the approxmatng functon (sold lne) versus frequency for the range khz to 0 khz. Fg. 1. Lnear-log plots of the measured phase of Z VC (crcles) and the approxmatng functon (sold lne) versus frequency. the phase response, some of whch are pronounced. Fg. 10. Lnear-log plots of the calculated phase of Z L (crcles) and the approxmatng functon (sold lne) versus frequency for the range khz to 0 khz. tons. Indeed, examnaton of the fgures shows better agreement n the hgh-frequency range where the mpedance of the nductor domnates. Examnaton of Fg. 1 shows a rsng asymptotc behavor n the hgh-frequency phase whereas Fg. (10) exhbts a constant phase. Ths dfference s caused by the phase of the motonal mpedance of the voce-col, whch s subtracted out n Fg. 10. Above the fundamental resonance frequency, the phase of the motonal mpedance s negatve, approachng zero as frequency s ncreased, thus causng the rsng behavor n the hgh-frequency phase n Fg. 1. The latter fgure shows the jog n the phase between 1.4 and 1.6 khz whch s not shown n Fg. 10. Ths s because the ponts chosen for Fg. 10 were n the range from khz to 0 khz. The author has had a great deal of experence wth student projects nvolvng loudspeaker measurements and has found the lossy nductor model descrbed here to gve excellent results. It s felt that the jog n the phase n Fg. 1 between 1.4 and 1.6 khz was due to a resonance effect n the daphragm and that t was responsble for the slght rpple observed n Fg. 10. The expermentaldatashownhereweremeasuredwthamlssa analyzer. A repeat of the measurements wth an Audo Precson System II analyzer showed an almost dentcal behavor. Smlar jogs n the measured phase have been observed wth many drvers. Indeed, some drvers exhbt multple jogs n VI. A SPICE MODEL AD SIMULATIO EXAMPLE A SPICE model for the lossy nductor s descrbed n ths secton and a smulaton of a closed-box woofer system s presented to llustrate the effect of the nductor losses on the frequency response of a drver. Fundamentals of SPICE smulatons of loudspeaker systems are covered n [1] and [8]. Fg. 13 shows a voltage controlled current source connected between nodes labeled 1 and. If the current through the source s equal to the voltage across t dvded by the mpedance of the lossy nductor gven by Eq. (19), then the source smulates the nductor. The analog behavoral modelng feature of PSpce can be used to mplement ths operaton wth the lne GZE 1 LAPLACE {V(1,)}={1/(K*PWR(S,n))} where numercal values must be used for K and n. Fg. 13. SPICE model for the lossy nductor.

7 448 JOURAL OF THE AUDIO EGIEERIG SOCIETY, VOL. 50, O. 6, 00 JUE The SPICE crcut for the smulaton s shown n Fg. 14. A 15 nch drver n a foot 3 box havng a Butterworth algnment wth a lower 3 db cutoff frequency of 40 Hz s assumed. The drver parameters are: voce-col resstance R E =7Ω, motor product B =1.6 T m, daphragm mass M MD = 197 grams, suspenson resstance R MS =8.07 s/m, suspenson complance C MS = m/, and daphragm pston area S D = m. The box parameters are: acoustc mass M AB =5.77 kg/m 4, acoustc resstance R AB = 1780 s/m 5, and acoustc complance C AB = m 5 /. The resstor R AL s necessary to prevent a floatng node n SPICE. Itsvaluewaschosentobe s/m 5, whch s large enough to be consdered an open crcut n the frequency response calculatons. Ths resstor can be consdered to model ar leaks n the enclosure. The front ar-load mass parameters are M A1 =.13 kg/m 4, R A1 =540 s/m 5, R A = 5760 s/m 5,andC A1 = m 5 /. TABLE I SPICE DECK FOR THE CLOSED BOX SIMULATIO CLOSED-BOX SIMULATIO *ACOUSTICAL CIRCUIT *ELECTRICAL CIRCUIT FSDUD VD VEG 1 0 AC 1V +707E-4 RE 1 7 LMA GZE 3 LAPLACE {V(,3)} RA ={1/(K*PWR(S,n))} RA HBLUD 3 4 VD 1.6 CA E-6 VD1 4 0 AC 0V LMAB *MECHAICAL CIRCUIT RAB HBLIC 5 0 VD1 1.6 CAB E-9 LMMD RAL E6 RMS VD3 1 0 AC 0V CMS E-6.AC DEC E4 ESDPS E-4.PROBE VD 9 0 AC 0V.ED Fg. 14. SPICE crcut for the example smulaton. The SPICE deck for the smulaton s gven n Table I. The lossy nductor s modeled by the voltage-controlled current source GZE between nodes and 3. The current through GZE,.e. through L E, s gven by the voltage V(,3) dvded by K*PWR(S,n),whereS s the complex frequency and numercal values must be suppled for K and n. The + sgn on the lne below GZE ndcates a contnued lne. Fve values of n between 1.0 and 0.5 were chosen for the smulatons. The values are n =1, 0.875, 0.75, 0.65, and0.5. Forn =1,thevalue K =0.001 was used, correspondng to a lossless nductor wth a value of 1 mh. For the other values of n, K was computed so that the SPL curves ntersect at the frequency for whch R E + K (jω) n = 3R E. Ths choce was made because t results n an ntersecton pont at a frequency that s approxmately the geometrc mean of 1 khz and 10 khz,.e. mdway betweenthesefrequencesonalogscale. TheK values are K =0.001, 0.003, , ,and The on-axs SPL at 1 meter s gven by [1] µ ρ0 fu D SPL =0log (31) where ρ 0 = 1.18 kg/m 3 s the densty of ar, f s the frequency, U D s the volume velocty output from the daphragm, and p ref = 10 5 Pa s the reference pressure. The SPL can p ref be dsplayed n the PROBE graphcs routne of PSpce wth the lne [1] 0*LOG10(59E3*FREQUECY*I(VD3)) where I(VD3) s the current through the voltage source VD3, whch s analogous to the volume velocty U D. Smulatons of the on-axs SPL at 1 meter are shownn Fg. 15. The plots show that the flattest overall response s obtaned wth the lossless nductor,.e. the curve labeled n =1. As the losses ncrease, the flat mdband regon dsappears and the curves become depressed above the fundamental resonance frequency. It s temptng to conclude from ths fgure that the nductor losses should be mnmzed for the best response. However, the curves for the lower values of n are calculated for a larger value of K. IfK s not ncreased as n s decreased, the nductor mpedance decreases wth n and the wdth of the flat mdband regon ncreases. Fg. 15. Smulated SPL responses for fve values of n and K. Correspondng plots for the magntude of the voce-col mpedance are shown n Fg. 16. Ths s dsplayed n the PROBE graphcs routne of PSpce wth the lne 1/I(VD1),.e. the source voltage (1 V) dvded by the current through the voltage source VD1, whch s the voce-col current.

8 LOUDSPEAKER VOICE-COIL IDUCTACE LOSSES 449 Fg. 16. Smulated voce-col mpedances for fve values of n and K. The author knows of no general relaton between K and n so that t s mpossble, n general, to predct how changes n one parameter affect the other. However, t has been observed that drvers havng lower values of n usually have hgher values of K. VII. COCLUSIOS It has been shown and expermentally demonstrated that the lossy nductance of a loudspeaker voce col can be modeled by two parameters K and n. The magntude of the mpedance vares as Kω n and the phase s nπ/. The parameters K and n can be determned from a lnear regresson analyss of measured voce-col mpedance data. In the author s experence, typcal observed values of n le n the range from 0.6 to 0.7. Changes n the magnet structure of a drver can affect both K and n. Future research for a general analytcal relaton between these parameters for typcal magnet structures could be of value n loudspeaker drver desgn. REFERECES [1] W.M.Leach,Jr.,Introducton to Electroacoustcs and Audo Amplfer Desgn, Second Edton, Revsed Prntng, (Kendall/Hunt, Dubuque, IA, 001). [] A.. Thele, Loudspeakers n Vented Boxes, Parts I and II, J. Audo Eng. Soc., vol. 19, pp (1971 May); pp (1971 June). [3] J. Vanderkooy, A Model of Loudspeaker Impedance Incorporatng Eddy Currents n the Pole Structure, J. Audo Eng. Soc., vol. 37, pp (1989 March). [4] J. R. Wrght, An Emprcal Model for Loudspeaker Motor Impedance, J. Audo Eng. Soc., vol. 38, pp (1990 Oct.). [5] H.W.Bode,etwork Analyss and Feedback Amplfer Desgn, (D.Van ostrand, Y, 1945). [6] A. E. Ftzgerald and C. Kngsley, Electrc Machnery, (McGraw-Hll, Y, 195). [7] R. H. Small, Drect-Radator Loudspeaker System Analyss, J. Audo Eng. Soc., vol. 0, pp , (197 June). [8] W. M. Leach, Jr., Computer-Aded Electroacoustc Desgn wth SPICE, J. Audo Eng. Soc., vol. 39, pp , (1991 July/Aug.).