Loudspeaker crossover networks

Transcription

1 oudpeaker croover network oudpeaker croover network By Tore A. Nielen Student 495 at TU, the Technical Univerity of enmark Augut 5 Abtract oudpeaker ytem ue croover network directing low and high frequencie to individual loudpeaker unit optimied for limited frequency range. The introduction of a croover network hould not degrade the reultant performance but the loudpeaker are phyically eparated, which introduce problem around the croover frequency when litening off-axi, and the individual repone of the loudpeaker unit further complicate ummation of the output ignal. The front baffle introduce reflection from the edge and the litening room add reflection from it boundarie cauing interference with the direct ignal eriouly affecting the reultant repone of the loudpeaker ytem. The objective of thi report i the tudy of croover network and the different caue that degrade the performance. Ørted TU Acoutical Technology

2 oudpeaker croover network Content. Introduction Croover network Threhold of hearing Audible range Change of level Group delay Muical intrument Cut-off lope Tranfer function Ideal filter Contant voltage filter Non-ideal filter All-pa filter Butterworth filter.... Croover network Firt-order Symmetrical two way Uing ba loudpeaker roll-off Second-order Aymmetrical two way Symmetrical two way Symmetrical three way Steep cut-off two way Third order Aymmetrical two way Symmetrical two way Symmetrical three way Steep cut-off two way Fourth order Symmetrical three way Steep cut-off two way Paive network Firt order Second order Third order Fourth order oudpeaker impedance Active network Firt order Second order igher order Special Model lectro-acoutical model The loudpeaker unit lectrical circuit Mechanical circuit Acoutical circuit... 5 Ørted TU Acoutical Technology

3 oudpeaker croover network..5. iaphragm velocity Sound preure oudpeaker pa band Sound preure level iaphragm excurion SPIC imulation model irectivity iffraction Circular baffle Sectional baffle Square baffle itening angle Two loudpeaker Three loudpeaker Boundary reflection One reflecting urface ectangular room ome entertainment Public addre oudpeaker characteritic Group delay Calculation method Implementation in MATAB Verification Aembling the model oudpeaker model Croover network Angular repone eflection Concluion eference Book Paper ink Appendix Plot tranfer function Main cript Filter function oudpeaker irectivity iffraction Boundary reflection Plot boundary reflection Ørted TU Acoutical Technology

4 oudpeaker croover network Foreword The current project wa initiated a a three-week coure to be executed in Augut of the 5 ummer vacation ince I could not participate in the normal three-week period in June. My profeor Finn Agerkvit accepted the propoal of a project to tudy croover network. The main objective wa the deign of croover network realiing a tranfer function of unity, i.e. flat amplitude and zero phae, and I planned to include the effect of loudpeaker bandwidth, the problem aociated with off-axi litening due to the diplacement of the loudpeaker on the front baffle and the interference from reflection within the litening room. Finn Agerkvit uggeted that I alo included the reflection due to diffraction. Initially I planned to ue SPIC for imulation and a pread heet for calculation, but I oon realied that it wa more appropriate to bae the imulation and calculation on MATAB. I decided to work through the loudpeaker model preented by each in order to derive a ueful model for loudpeaker, and I included the effect of the voice coil inductance and combined the low and high frequency model from each into one ingle model, which cover the frequency range below diaphragm break-up. A the project progreed, I realied the need to include group delay and it eemed appropriate to add note on the threhold of hearing thu defining an acceptance limit for ue during the development of a croover network. According to my log, I have been working for 8 hour, which i 5 % more than the nominal workload for a three-week coure. If an unlimited amount of time were available, I would have worked more on high-order croover filter and improved the ection on diffraction, off-axi litening, boundary reflection and group delay. Tore A. Nielen Augut 4, 5. Ørted TU Acoutical Technology 4

5 oudpeaker croover network. Introduction Ideally, a ingle loudpeaker hould reproduce the full audible frequency range without any detectable ditortion, but thi i unfortunately not poible although good full-range loudpeaker do exit. The frequency range of a full-range loudpeaker i limited with weak ba and unatifactory treble, the frequency repone i irregular or at leat compromied by the directivity at high frequencie and it i difficult to keep ditortion low when the ame diaphragm i ued for ba and treble. ow frequencie move the diaphragm ignificantly at high ound preure level thu introducing harmonic ditortion related to loudpeaker contruction (magnet, voice coil and upenion) and inter-modulation between ba and treble caued by the opplereffect. The one and only way of ditortion reduction i decreaing diaphragm excurion, but thi require an increae of diaphragm area to compenate for the lot ound preure; and enlarging loudpeaker ize woren high frequency reproduction. It all boil down to a requirement of loudpeaker optimied for reproduction of a limited frequency range and thu the need of a frequency dividing network... Croover network A pair of typical croover network are hown in Figure. To the left i a two-way ytem, which could ue a croover frequency around kz, and to the right i a threeway ytem, which could ue croover frequencie around 8 z and 4 kz. Audio ource PA igh-pa filter ow-pa filter Audio ource igh-pa filter Band-pa filter ow-pa filter PA PA PA igh-frequency loudpeaker (Treble) Mid-frequency loudpeaker (Midrange) ow-frequency loudpeaker (Ba) Figure ayout of a typical cro-over network, which can be paive, i.e. coniting of capacitor, inductor and reitor and driven from a ingle power amplifier (left), or it can be active, i.e. uing operational amplifier with frequency-dependent feedback and individual power amplifier for each channel (right). The loudpeaker are driven from power amplifier, which can either be located before the croover network; the conventional approach uing paive croover network, or the power amplifier can be located between the croover network and the loudpeaker; thu requiring an amplifier for each loudpeaker. The paive croover network i currently the mot ued approach but the active croover network i expected to be increaingly popular in the near future ince highquality power amplifier module baed upon the witch-mode technique (Cla ) are becoming a eriou alternative to the linear power amplifier of today. In addition to improved control of filter parameter and protection of the loudpeaker do the active croover network offer electrical control of the moving ytem parameter and adjutment of the repone to the litening room. But before entering the tudy of croover network, a few word on what can be heard, and what cannot, i required. Ørted TU Acoutical Technology 5

6 oudpeaker croover network.. Threhold of hearing There i no idea in optimiing a filter if the improvement i inaudible and the money could be pend better on other job. So, here i a brief overview of what i audible and what i not. on t take the limit too literately; they are meant a guideline.... Audible range The audible range i defined by the Fletcher-Munon curve reproduced to the left in Figure. They publihed their data in 9 uing headphone; meaurement uing anechoic chamber were publihed in 956 by obinon-adon and later reviewed and tandardied in a ISO 6, hown to the right. Figure qual loudne contour due to Fletcher-Munon ( and to obinon-adon ( We can hear ignal from 5 z to 5 kz but few loudpeaker ytem can reproduce thi range, at leat not at realitic level ince low-frequency ignal require large diaphragm and long voice coil in order to move the air; the ound preure level mut exceed 8 db at very low frequencie to be heard and db i required at z to balance a typical peech level around 65 db. Organ muic may extend to 6 z for organ fitted with feet pipe, but they are rarely found and rarely ued o organ muic i limited to z. Piano muic may extend to 7 z, while jazz, rock and popular muic eldom pae below 4 z; the lowet tring on the acoutical double ba and the electric ba guitar. Powerful lowfrequency ignal may, however, arie from electrical keyboard, computerized effect and recording of large drum, machine, thunder torm and exploion. No one can hear ound above kz, and aging further reduce the limit, o a pragmatic upper limit would be 5 kz. People with golden ear may potulate that the treble unit hould extend far beyond kz to avoid phae ditortion. The audibility of phae i controverial, o a afe view would be that reproduction beyond kz do not harm; but have in mind that FM-radio broadcating limit the range to 5 kz and C-recording to kz. Modern ignal tranmiion uing MPG and other format often ue the 44. kz ampling frequency of the C-media thu haring the limit.... Change of level The ability to detect a change of level i between.5 db and db [] o loudpeaker artefact below thi limit can be expected inaudible. Thi i quite fortunate, ince Ørted TU Acoutical Technology 6

7 oudpeaker croover network irregularitie of thi order of magnitude mut be expected with loudpeaker. It i obviou that the lowet limit hould be ued deigning high-end equipment, ince the litener can be expected to have trained ear.... Group delay Threhold of audibility of group delay i being debated, but it hould be relatively afe tating the limit a a couple of milliecond for ignal in the 5 z to 8 kz range. (Source: Muical intrument The fundamental note of muical intrument i typical located below kz a hown in Figure. The harmonic overtone extend beyond the hearing limit but the level i reduced toward the higher frequencie. The decay i trongly dependent upon the actual intrument being examined, but an average lope would be around 6 db/octave. Mot muic ue fundamental within the ½ octave range from the lower C-note at 65 z to the upper g-note at 78 z, o the muical power i mainly retricted to thi range. uman voice Bra enemble Wood wind String quartet Orcheter Piano Organ ynamic range 5 db 5 db 5 db 5 db 7 db 5 db 5 db z z kz kz Figure Frequency range for the fundamental note of muical intrument [5]. The dynamic range extend from 4 db to db for acoutical intrument and higher for electrically amplified muic. A ymphony or rock orchetra cannot be reproduced at realitic level for home entertainment, o the playback level mut typically be reduced. ecording level wa previouly compreed manually by increaing the weaket level during recording, in order to cut the mater dik (P record) keeping the weaket ignal above the noie floor of the medium. Thi i not needed nowaday for recording of compact dik (C), where the dynamic level i, at leat theoretically, 96 db. educing the reproduction level move the weaket ignal below the threhold of hearing, at leat for the lowet frequencie, o the loudpeaker ytem may require a ba boot for reproduction at reduced level. Thi i ometime called phyiological loudne contour and i included with many amplifier ytem. The ucce of thi correction i dependent upon the et-up of the combined ytem, coniting of the ource, the amplifier, the loudpeaker and the ize of the room and i mot effectively implemented with integrated ytem where the interconnection level are known. The correction i often accompanied by a treble boot a well, but thi i baed on a miinterpretation of the equal loudne contour; the treble part of the equal loudne contour i turned upward at high frequency thu indicating reduced enitivity at high frequencie, but the ditance between the different level i almot contant o correction i not required. Ørted TU Acoutical Technology 7

8 oudpeaker croover network.4. Cut-off lope A croover network i not a brick wall filter with infinite attenuation outide the pa band; the croover network gradually attenuate the ignal above or below the croover frequency a illutrated in Figure 4 for low-pa and high-pa filter. The pa band i the frequency range where attenuation i minimum, typically db although ome filter attenuate more than thi. The tranition band i the frequency range where the attenuation i becoming active and the top band i the frequency range where the attenuation ha become ufficient to efficiently remove the loudpeaker. Figure 4 Filter characteritic for Butterworth filter with order,,, 4 and 6. A loudpeaker contribute with audible output in the tranition band, which mut be taken into account to avoid interference between the loudpeaker. Sufficient amount of attenuation i obtained when the level from the attenuated channel i le than a certain limit, for intance below db. The limit define the acceptable interference level. Aume for example, that the loudpeaker peak 6 db at ome frequency near the top band and that thi mut be attenuated. With db of intended attenuation thi correpond an actual level of 4 db, or a ound preure of -4/ % of the nominal level, which may reult in ± db of interference. A croover network with a cut-off lope of ±6 db/octave indicate that the loudpeaker mut be well-behaved for at leat octave beyond the cro-over frequency. A ba loudpeaker, which i to be cut-out above kz, mut be reaonably flat to 6 kz. The croover network mut protect the treble loudpeaker from the high power level at lower frequencie. The loudpeaker i compliance-controlled below the reonance frequency, which i uually around kz, o the diaphragm move in proportion to the applied voltage. The low-frequency excurion of the diaphragm may be inaudible but it may give rie to audible ditortion of high-frequency ignal when the low-frequency excurion of the upenion ytem reache the non-linear region. The unneceary diipation of power heat the voice coil and may damage the treble loudpeaker, which i capable of handling few watt only. Mot of the ignal power in muic i located below approximately 5 z o a power reduction well below W of diipation within the treble loudpeaker require in exce of db of attenuation for ufficient protection. If the treble loudpeaker i to be cut-in at kz, which i jut two octave above 5 z, the required filter lope become a minimum of db/octave. Ørted TU Acoutical Technology 8

9 oudpeaker croover network.5. Tranfer function All channel of the croover network will be decribed by their tranfer function, which i a convenient way of decribing an electrical filter, and ytem analogie allow traightforward tranformation to mechanical and acoutical ytem a well. The tranfer function may be defined from requirement uch a flat amplitude, which can then be ued to pecify the ytem parameter. Input ignal IN Bandwidth-limited channel # Bandwidth-limited channel # OUT IN Σ OUT IN OUT ( ) IN Output ignal Figure 5 A croover network conit of two or more filter channel dividing the frequency range between the loudpeaker. The output from the loudpeaker are ummed at the obervation point; i.e. at the ear of the litener. Tranfer function will be expreed in the frequency domain where frequency i repreented by the aplace operator, defined by: α iω. The initial condition are decribed by α and ω πf i the angular frequency. Throughout thi report i α, o iω can be aumed although the formula are generally valid and may, if required, be tranformed to the time domain uing the invere aplace tranform. owever, time domain repreentation are not referenced in thi document. The complex frequency operator will be normalied by diviion with ω, which repreent the cut-off frequency in mot ituation. A tranfer function i defined by the excitation input to the network, IN, and the output repone from the network, OUT. Auming inuoidal excitation: OUT IN ω exp ( iωt) An input excitation ignal IN i routed in parallel to the channel of a network with the individual tranfer function,,... and the output ignal become: OUT IN, OUT IN The loudpeaker will, for the moment, be conidered ideal, o the acoutical output of the loudpeaker i a true copy of the electrical input ignal. Auming linearity, the ignal from the individual channel are added at the receiver: OUT OUT OUT K Uing the above definition of the tranfer function, the um can be written: OUT ( M ) IN K The um of the individual tranfer function i defined a the ytem tranfer function. K M,... Ørted TU Acoutical Technology 9

10 oudpeaker croover network The main objective of thi report i analying the ytem tranfer function for the complete ytem involving the croover network, the loudpeaker, the front baffle and the litening room. In order to do o, a reference tranfer function i needed for comparion. It i obviou, that an ideal tranfer function hould not add or remove any information, it hould be a tring of wire. A caling factor, different from unity, i allowed, ince amplification, attenuation, ign inverion or time delay within the ytem i not conidered a ditortion of the ignal. The caling factor may alo include a dimenion for tranformation between the electrical, mechanical or acoutical ytem..5.. Ideal filter Contant voltage filter Although loudpeaker are far from ideal, avoiding approximation in the deign of the croover network it i a good tarting point. The deign of croover network fulfilling the requirement, which guarantee flat amplitude and a phae of zero, are called contant-voltage filter and will be baed upon the following polynomial: N N PN a a an The contant a, a,, a N- define the filter type, N i the order of the polynomial and the coefficient are hown for the Butterworth filter type later in thi chapter. Other polynomial, uch a Beel or Chebychev, can ue the above polynomial form if they are normalied to unity for coefficient a and a N. If thi i not the cae, all term of the polynomial mut be divided by a, and the normaliation coefficient ω mut thereafter be corrected to include a and a N. An alternative repreentation of P N i the product form, which i uing the root of the polynomial. The third order Butterworth polynomial can be repreented by the following two identical expreion: P P ( ) (.5 i.866 ) (.5 i.866 ) After the multiplication are carried out the original polynomial reult. A tranfer function of order N can now be defined: Q b b N N N N N PN a a an b The tranfer function atify the requirement when a i b i ince the nominator polynomial Q N and denominator polynomial P N are identical, but the requirement will be violated, if a i b i for one or more of the term. Thi violation may be required building croover network of high order where the ideal tranfer function become cumberome. The reult i anyway a valid tranfer function although the phae repone and poibly alo the amplitude repone will be affected. N o N o Ørted TU Acoutical Technology

11 oudpeaker croover network iviion into croover channel ue the following identity: b b N N N N PN PN PN PN PN ach term repreent a tranfer function with it own characteritic and two or more term can be combined a required. All term hare the ame denominator polynomial and are thu of the ame order, regardle of the actual order of the nominator. Two example will introduce the method. xample. For a firt-order croover only two term are available o the deign leave no choice other than accepting the following arrangement: P P N P b P xample. For a croover network of ufficiently high order (N 4), the firt two term of P N can be ued for the ba channel, the lat two term for the treble channel and the remaining term are available for the midrange channel: PN N N b b bn bn BPN PN P P P N The cut-off lope are proportional to /f N- for the ba channel, f N- for the treble channel and f and /f for the midrange channel. A fourth order filter would offer cut-off lope of ±8 db/octave for ba and treble and ± db/octave for midrange. Both filter are realied in the next chapter together with other implementation..5.. Non-ideal filter All-pa filter The above method i ueful for filter up to fourth order but become cumberome for higher order. A olution i to remove all middle term and ue only the firt and lat term repreenting the low-pa and high-pa channel. Thee filter, which are called all-pa filter, can be ued for croover network of any order. The requirement i not atified o the phae of the filter will be different from zero. With b, b,, b N- et to zero the tranfer function become: N N PN and PN N PN PN PN The firt term repreent a low-pa channel with a cut-off frequency of f ω /π and a cut-off lope of /f N, o the higher frequencie are attenuated by 6N db/octave. Filter amplitude repone i flat under certain condition, which will be analyed below. Firtorder filter are ideal (contant-voltage filter) and will not need pecial conideration. PN P P N N N N Amplitude The econd term repreent a high-pa channel with a cut-off frequency of f ω /π and a cut-off lope of f N, o the lower frequencie are attenuated by 6N db/octave. f N N -6N db/octave Frequency Ørted TU Acoutical Technology

12 oudpeaker croover network PN P P N N N N N N N When the output from both channel are combined the following tranfer function for the complete ytem i obtained: N N P N At croover : N i P At croover ( i) i the um depending upon the filter order and for N, 6,,... the term cancel no ignal i being tranmitted at croover. The cancellation can be avoided by ign inverion one of the channel, which ha the conequence that the phae move from to 8 through the frequency range. For N odd are the channel 9 or 7 out of phae, and the channel combine at croover with db of lo. To avoid peaking, the tranfer function mut be deigned to db at the croover frequency and thi can be realied uing a Butterworth polynomial a bai for the deign. For N even are two channel in-phae if the above mentioned ign inverion i included a required and the channel combine at croover without lo. To avoid peaking, the tranfer function mut be deigned to 6 db at the croover frequency. Thi can be realied uing a quared Butterworth polynomial a bai for the deign; a deign method referred to a the inkwitz-iley filter deign. Contant amplitude, N, i realied by the o-called all-pa filter, which are baed upon the Butterworth polynomial. Firt-order croover network are born a ideal filter (i.e. contant voltage), o the amplitude i contant. Thi will be demontrated below uing variable w ω/ω a a ubtitute for in order to identify real and imaginary part of the frequency variable. The all-pa variant of the firt-order croover network ( in the nominator) i alo ueful and i analyed a well []. ± ± iw iw Filter of higher order can realie the all-pa function when the polynomial ued i of Butterworth characteritic, ince thi allow factoriation of the nominator and denominator polynomial. Second-order croover network require inverion of the high-pa channel to avoid the notch filter and the level mut be 6 db at croover, o the croover network i baed on a quared firt-order Butterworth polynomial. The amplitude repone become []: ( )( ) ( )( ) w w Amplitude iw iw ( ) w 6N db/octave N N f w Frequency Ørted TU Acoutical Technology

13 oudpeaker croover network Ørted TU Acoutical Technology Third-order croover network are inenitive to the ign of the treble channel (the amplitude repone i unaltered although the phae repone i changed). The thirdorder Butterworth polynomial ue a a, and the factoriation reult in []: ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) w w iw iw w w w w w iw w iw Fourth-order croover network require a level of 6 db at croover and can be baed upon a quared econd-order Butterworth polynomial (a ). The amplitude repone become []: ( ) ( )( ) ( ) ( ) ( ) 4 4 w w w w i w w i igher order filter are not covered in thi report but can, according to reference [], be hown to fulfil the all-pa filter requirement..5.. Butterworth filter Croover network are often baed upon the Butterworth polynomial ince thi lead to good all-pa filter. The amplitude repone for a Butterworth low-pa filter i [5]: N N N PN ω ω ω ω The amplitude approache unity for zero frequency, the amplitude i / or db at the croover frequency regardle of the filter order, and the aymptotic cut-off lope for higher frequencie i 6N db/octave. Thi i a maximally flat filter and one of it characteritic i, that all filter block of the phyical implementation will be deigned for the ame cut-off frequency but with different quality factor. Other filter characteritic are realied through frequency tranformation, introduced briefly below [5], but the tranformation are not required for the current analyi ince the different channel are generated directly from the tranfer function and not from tranformation of a model filter.

14 oudpeaker croover network A high-pa filter i realied by tranformation of into / : PN N ω ω N ω ω A band-pa filter i realied by the following tranformation, which conit of / a well a in combination with a coefficient B, repreenting the bandwidth of the reulting filter: B BPN B N B i N B N B N B N N ω ω N ω ω The coefficient and root of the Butterworth polynomial i hown in the table below for filter from econd to eventh order. The firt-order polynomial i Butterworth too but do not include coefficient to be defined. Table Coefficient and root of Butterworth polynomial (from [5]). Order Coefficient oot a.44 (-.77 ±i.77) a a. (-) (-.5 ±i.866) 4 a a.6 (-.87 ±i.99) a.44 (-.99 ±i.87) 5 (-) a a 4.6 (-.9 ±i.95) a a 5.6 (-.89 ±i.5878) 6 7 a a a a a 9.46 a a a a a a (-.588 ±i.9659) (-.77 ±i.77) ( ±i.49) (-) (-.5 ±i.9749) (-.65 ±i.788) (-.9 ±i.49) N Ørted TU Acoutical Technology 4

15 oudpeaker croover network. Croover network Ideal filter, defined by the tranfer function requirement, were expected to realie good croover network but thi proved to be the cae only for low-order network; higher order network tend to include term complicating the contruction and obtructing the operation. All-pa filter, realiing two-way ytem with maximum cut-off lope, were alo analyed and proved to operate a expected... Firt-order Croover network of firt order are popular becaue the component cot i low; it may ometime be poible deigning croover network with one ingle capacitor for the treble loudpeaker. owever, the cut-off lope i ±6 db/octave, which i inufficient in mot cae ince the loudpeaker mut reproduce well octave pat the croover frequency. Thi i problematic ince ba loudpeaker uffer from increaed directivity and diaphragm break-up at high frequencie, which concentrate the ound on-axi and typically generate a ragged high-frequency repone, and treble loudpeaker cannot reproduce below the reonance frequency. In addition to thi i the filter inufficient in protecting the treble loudpeaker againt detructive low-frequency ignal, o the firt-order croover i limited to loudpeaker intended for low ound preure level and i typically found in low-cot deign. A firt-order tranfer function i defined from N with N :... Symmetrical two way The equation i eparated into two channel with a low-pa channel for the ba loudpeaker and a high-pa channel for the treble loudpeaker. P P ealiation i imple, only one component i required for each branch but the filter i very dependent upon the impedance of the loudpeaker o the paive network may not prove atifactory in real life. B CT Ba B Treble T Figure 6 Paive croover network. The attenuation i a expected only when the impedance of the loudpeaker are contant, o impedance compenation i required in mot cae hence oppoing the implicity of the deign. Ørted TU Acoutical Technology 5

16 oudpeaker croover network The reult i hown below. The channel add up to unity and the phae to zero, a they hould ince thi i an ideal filter (contant-voltage) and auming perfect loudpeaker with identical path length to the litener. Figure 7 Amplitude and phae repone. The loudpeaker are 9 out of phae throughout frequency, thu adding with db of lo when combined. The channel are at db at croover and add to db. A variant of the filter i realied by inverting the treble loudpeaker, which generate the following tranfer function: Thi i an all-pa filter, where the amplitude i flat but the phae i changing gradually from to 8 through frequency with 9 at croover. Figure 8 Amplitude and phae repone with the treble loudpeaker inverted. The loudpeaker are 9 out of phae throughout frequency. The introduction of a phae different from zero for the complete croover network introduce a group delay different from zero and thi i hown in Figure 9. Note that the group delay unit i calculated for a normalied filter correponding to a cut-off frequency of z. With a cut-off frequency of z the group delay caling will be in milliecond and not econd. Ørted TU Acoutical Technology 6

17 oudpeaker croover network Figure 9 Group delay with the treble loudpeaker inverted.... Uing ba loudpeaker roll-off Some ba loudpeaker are deigned to roll off moothly above a certain frequency, typically in the range where the croover frequency i placed. The croover network can be implified if thi roll off i ued a the low-pa filter thu only implementing the high-pa filter capacitor for the treble loudpeaker. Thi reult in a low-cot croover network ince only one capacitor i required. The ba loudpeaker mut realie the required tranfer function, o the db frequency of the ba loudpeaker dictate the croover frequency. CT Ba Treble Figure Paive croover network uing the ba loudpeaker roll off. Treble loudpeaker build from piezoelectric tranducer with an integrated horn are available with a cut-off frequency of approximately 4 kz and can be ued without external component. The treble loudpeaker accept up to V applied directly and can be ued for a two-way ytem without any component within the croover network but the cut-off i very harp o the reulting deign i not a firt-order croover network. It i poible to electrically adjut the cut-off frequency of the ba loudpeaker uing pole-zero compenation, which i mot effectively implemented uing active filtering. The method can be ued to move the ba loudpeaker voice coil cut-off frequency from the actual value to the deired value. Aume that the ba loudpeaker cut-off frequency i at ω, and not at ω a required. The tranfer function can be rewritten to include a null and a pole at the new frequency. The ratio of the new null/pole tranfer function i unity o the tranfer function i not changed. P CN S ( )( ) Ørted TU Acoutical Technology 7

18 oudpeaker croover network The term are then arranged o the zero i moved to the croover network CN. The tranfer function for the croover network CN and the ba loudpeaker S then become: CN S The croover network CN i now a correction filter, which modifie the amplitude pectrum of the low-frequency channel making the ba loudpeaker ueful with the required croover frequency. The correction hould not be brought too far, however, but minor correction of the order of ±6 db (one octave up or down) hould be realiable. Note, that moving the croover frequency upward require amplification of the ignal fed to the ba loudpeaker and moving the croover frequency down require attenuation of the ignal fed to the ba loudpeaker. The former i impoible to implement uing paive filter and the latter impractical, hence the recommendation of active filtering. Ørted TU Acoutical Technology 8

19 oudpeaker croover network.. Second-order Croover network of econd order are popular due to the low component count and relatively teep cut-off lope. In addition i the deigner provided with an intereting collection of filter to elect among; both the ideal contant-voltage and the non-ideal all-pa filter are offered. A econd-order tranfer function i defined from N with N : a a Coefficient a define the filter characteritic around the croover frequency and i conventionally defined by the quality factor Q of a econd-order circuitry: a A common quality factor i.7 for the Butterworth characteritic, which i db at the croover frequency, but any value can be ued with Q.5 to a typical value.... Aymmetrical two way One obviou realiation of a two-way croover network i to divide between the firtorder and econd-order term thu increaing the cut-off lope for the treble loudpeaker to improve the protection. P P Q a a a The high-pa filter i of econd-order with a lope of db/octave, which i ufficient to protect the treble loudpeaker. A ueful range of.5 octave below the cut-off frequency i required for db of attenuation o the croover frequency mut be higher than.5.8 time the reonance frequency of the treble loudpeaker. At the croover frequency, and auming a, we get: P P i i i i i i i.5 9 i. 7 So, the channel are 7 apart and will add with ome lo, hence the light boot of the low-frequency channel at the croover frequency. For a both channel are booted but the um remain contant at unity o the phae of the channel are moved further apart for reduced value of a. Thi i not a good way of deigning a croover network; the channel hould not oppoe each other ince the reult then become a difference between two fighting channel. A ound way of deigning i to elect a fairly large value of the coefficient, where a eem to be a fair compromie. Ørted TU Acoutical Technology 9

20 oudpeaker croover network B CT B Ba B T Treble T CB Figure Paive croover network. Active realiation: P P The tranfer function P i implemented a a tandard econd-order high-pa filter with Q /a, and the low-pa channel can be derived by an operational amplifier. A value of a reult in Q, which i a db Chebychev characteritic with relative teep cut-off and thi alo applie to the low-pa channel, which i cut-off after one decade. A value of a reult in Q.5, which i the limit where the root of the polynomial become real. Thi remove any tendency to ocillate in the treble channel, hence the mooth tranition without peaking. The cut-off of the low-pa channel i omewhat weaker o the ba loudpeaker mut operate well to at leat twenty time the cut-off frequency. Figure Amplitude repone with a (left) and a (right). The quality factor refer to the treble channel and i Q (left) and Q.5 (right). The low-pa filter i econd-order but the firt-order term in the nominator reduce the cut-off lope to 6 db/octave at high frequencie, which require a ba loudpeaker capable of operating octave above the cut-off frequency, o the channel i more or le full range and the ba loudpeaker mut perform well at high frequencie. Ørted TU Acoutical Technology

21 oudpeaker croover network... Symmetrical two way The firt-order term of the tranfer function can be plit into two halve o both channel include a firt-order term. P P a a a a Both filter are of econd-order but the lope i ±6 db/octave, which i inufficient to protect the treble loudpeaker o the olution hould not be ued for high-power ytem. The loudpeaker mut be capable of operating octave outide the croover frequency At the croover frequency, and auming a, we get: P P i i.7 45 i i i i.7 45 i i So, the channel are 9 apart at croover and will add with db lo to db. educing the coefficient to a introduce peaking in both channel to compenate for the increaed phae difference. B CT B T Ba Treble CB B T T Figure Paive croover network. The coefficient adjut the behaviour of the filter and two example are hown below. The deign with a reult in teep cut-off but there i a tendency for ringing on tranient although the two channel will cancel when combined. It could be taken a a warning for problem with off-axi litening where the output from the ba loudpeaker i reduced due to it directivity. Ørted TU Acoutical Technology

22 oudpeaker croover network Figure 4 Amplitude repone with a (left) and a (right). It appear that a hould be in the range from.5 to a a tarting point at leat. A very mooth reult i obtained with a value of.6, which i hown in Figure 5. Figure 5 Amplitude repone with a Symmetrical three way The equation can be plit into three channel: P BP P a a a a The cut-off lope i ± db/octave for the ba and treble channel and ±6 db/octave for the midrange channel o the treble loudpeaker i protected but the midrange loudpeaker mut cover a range of 6 octave total. Coefficient a repreent the quality factor (Q / a ), thu defining the pule repone of the individual channel; the reultant pule repone of the complete ytem i unity when the output are combined, auming perfect addition of the channel. Ørted TU Acoutical Technology

23 oudpeaker croover network At the croover frequency, and auming a, we get: P P i i i BP i i i So, the low-pa and high-pa channel cancel at croover and leave the midrange to fill the gap. The deign wa originally propoed by Bang & Olufen and labelled a the Filler river ytem. The name indicate that the middle channel wa not conidered a conventional midrange channel but rather a phae correction of a two-way ytem. B CM CT M Ba Treble Treble CB B T T T Figure 6 Paive croover network. Two example are hown, uing Q, which repreent a db Chebychev filter characteritic of the econd-order filter, and Q.5, which repreent the limit where the channel are unconditionally table (real root). Figure 7 Amplitude and phae repone with a. The phae difference between the neighbour channel i 9 throughout frequency and 8 between the low-pa and high-pa channel. Ørted TU Acoutical Technology

24 oudpeaker croover network Ørted TU Acoutical Technology 4 Figure 8 Amplitude and phae repone with a...4. Steep cut-off two way Maximum cut-off lope of both channel i obtained if the firt-order term i removed from the nominator, reulting in a two-way croover. The filter doe not atify the requirement of unity tranfer function ince but it can realie an all-pa filter. a a P P etermination of the coefficient aume modelling by a combination of two firt-order Butterworth filter in cacade (the inkwitz-iley method). The nominator polynomial i not important for thi evaluation, only the denominator polynomial i conidered. N N P P By comparion, the coefficient i found to: a The tranfer function of the channel become: o P o P and The reultant tranfer function become: o At croover i i o the nominator equate zero; i.e. the filter introduce a notch at the croover frequency a can be een from Figure.

25 oudpeaker croover network B CT CB Ba B T Treble T Figure 9 Paive croover network. Figure Amplitude phae repone with a. The notch at croover i due to the phae difference between the loudpeaker. The ba loudpeaker i 9 and the treble loudpeaker i 9, i.e. 8 apart, o the output cancel. A olution i to invert the polarity of one of the channel, often the treble loudpeaker, which retore the phae difference to. The reultant tranfer function become: P P a The reulting amplitude repone i flat but the phae decreae gradually from at low frequencie to 8 at high frequencie. Thi i a mall price to pay for a filter with ufficient cut-off lope to reduce the bandwidth requirement to octave outide cut-off and to protect the treble loudpeaker againt low-frequency ignal. Figure Amplitude and phae repone with a and inverted treble channel. Ørted TU Acoutical Technology 5

26 oudpeaker croover network The introduction of a phae different from zero introduce a group delay, which i hown in Figure. Note that the group delay unit i calculated for a normalied filter correponding to a cut-off frequency of z. With a cut-off frequency of z the group delay caling will be in milliecond and not econd. Figure Group delay with a and inverted treble channel. The filter i very popular due to the low component count of two component per channel and relative teep cut-off. Ørted TU Acoutical Technology 6

27 oudpeaker croover network.. Third order Ideal croover network of third order are problematic, a will be hown in thi ection. The problem being large phae difference between channel, which reult in quite odd deign. A third-order tranfer function i defined from N with N : a a a a etermination of the coefficient aume modelling by a combination of a firt-order filter and a econd-order filter in cacade. The nominator polynomial i not important for thi evaluation, only the denominator polynomial i conidered. P P P By comparion, the coefficient are found to: N N c a c a N N ( c) ( c) c The value of c mut be choen for the ue with all-pa filter. The phae difference between the channel i 7 at croover (ince i ), o the ignal are added with a lo of db. The croover filter hould thu be db and ince the firt-order filter realie thi, the econd-order filter mut be et to db at croover o it require c ince c /Q for the econd-order filter define the level at reonance (equal to Q).... Aymmetrical two way One obviou realiation of a two-way croover network i to divide between the firtorder and econd-order term: P P a a a a a a A paive implementation i enitive to loudpeaker impedance becaue of the erie element. The damping mut be upplied by the load reitance, the loudpeaker, which i far from reitive if not compenated properly, o the paive croover network require impedance compenation of both branche. Ørted TU Acoutical Technology 7

28 oudpeaker croover network At croover, and auming a a for implicity, we get: P P i i i.7 5 i i i i i i i.58 8 i i i i So, the channel are 5 out of phae at croover but the level i increaed for the low-frequency channel to compenate for the lo and the channel add up to. Thi i not a healthy way of deigning a croover network, the deign hould not be baed upon ubtraction of large figure; thi will eaily lead to problem. B CT B CT B B Ba B T Treble T CB Figure Paive croover network. An active filter realiation could ue the following algorithm to extract the low-pa channel from the high-pa channel. P P The reulting amplitude repone i hown below for two arbitrarily elected value of the coefficient a and a. Figure 4 Amplitude repone with a a (left) and a a (right). The peaking become wore for maller value and the lope become too oft for larger value of a and a o thi i not a particularly valuable deign. The high-pa filter cutoff lope i 8 db/octave, but the low-pa channel i only 6 db/octave o the loudpeaker mut be well-behaved three octave above the cut-off frequency. Ørted TU Acoutical Technology 8

29 oudpeaker croover network... Symmetrical two way One obviou realiation of a two-way croover network i to divide between the econd-order and third-order term: P P a a a a a a Cut-off lope i ± db for both channel and the filter i ymmetrical for a a. At croover, and auming a a for implicity, we get: P P i i i i i i i i i i i i i i So, the channel are 44 out of phae at croover but the level i increaed for both channel to compenate for the lo and the channel add up to. Thi i not a healthy way of deigning a croover network, the deign hould not be baed upon ubtraction of large figure; thi will eaily lead to problem. A paive implementation i enitive to loudpeaker impedance becaue of the erie element. The damping mut be upplied by the load reitance, the loudpeaker, which i far from reitive if not compenated properly. B CT B CT B B CB Ba B T Treble T Figure 5 Paive croover network. An active filter realiation could ue the following algorithm to extract the low-pa channel from the high-pa channel. P P The reulting amplitude repone i hown below for two arbitrarily elected value of the coefficient a and a. The cut-off lope i ufficient to reduce the loudpeaker requirement to octave pat the croover frequency and uing a value of a a around 4 eem ueful ince the peaking i limited to around db for each channel, which could be expected to work in real life. Ørted TU Acoutical Technology 9

30 oudpeaker croover network Ørted TU Acoutical Technology Figure 6 Amplitude repone with a a (left) and a a.7 (right).... Symmetrical three way A ymmetrical three-way croover network can be build by uing the middle two term for the midrange loudpeaker: a a a a a a a a P BP P Cut-off lope i ±8 db/octave for the ba and treble channel but only ±6 db/octave for the midrange channel. The filter will be ymmetrical for a a. A paive implementation i not attractive but the active olution i traightforward, when the low-pa and high-pa channel have been contructed: P P BP The low-pa and high-pa channel can be contructed from tandard third-order Butterworth filter block but the deign i quite tricky a the following analyi will how. Aume that the coefficient are a a, to implify the analyi. The following amplitude and phae can then be found at the croover frequency ( i): i i i i i i i i i i i i i i i i i i P BP P So, the low-pa and high-pa channel combine to at croover while the midrange channel i at, which mean that the two channel oppoe the midrange channel. The midrange channel mut be booted to in order for the um of all channel to be unity.

31 oudpeaker croover network Figure 7 - Amplitude repone for the ymmetrical third-order three-way croover network with a a.5 (left) and a a 4 (right). The concluion i, that the deign would do better without the midrange channel, and thi i exactly the following croover network to be analyed. Thi i at the end of the ideal filter with, ince higher order filter include too many term; they are cumberome to implement, epecially with paive filter...4. Steep cut-off two way In order to improve the cut-off lope of the low-pa channel one method i to remove the a and a coefficient of the nominator: P P a a a a An active filter realiation could ue the following algorithm to extract the low-pa channel from the high-pa channel. P P Again, the paive filter mut ue impedance compenation of the loudpeaker. B CT B CT CB Ba B T Treble T Figure 8 Paive croover network. The coefficient mut be a a for the all-pa filter and the reult i hown in Figure 9. The phae doe not jump 6 at croover; thi i due to the MATAB angle function. Ørted TU Acoutical Technology

32 oudpeaker croover network Figure 9 - Amplitude and phae repone with a a. The introduction of a phae different from zero introduce a group delay hown in Figure. Note that the group delay unit i calculated for a normalied filter correponding to a cut-off frequency of z. With a cut-off frequency of z the group delay caling will be in milliecond and not econd. Figure Group delay with a a. The filter include two coefficient and it enitivity to variation wa analyed by caling the coefficient by ± % with the reult hown in Figure. Ørted TU Acoutical Technology

33 oudpeaker croover network Figure - Amplitude repone for % change of coefficient: a.8, a. (left) and a., a.8 (right). The filter accept inverion of the treble loudpeaker. Figure - Amplitude and phae repone with a a and inverted treble. The group delay i reduced in amplitude and become monotonically a reult of the inverion, o although one may argument againt the inverion, there could be an audible improvement by doing o. Figure Group delay with a a and inverted treble loudpeaker. Ørted TU Acoutical Technology

34 oudpeaker croover network Ørted TU Acoutical Technology 4.4. Fourth order Croover network of fourth-order are popular due to the teep cut-off lope, which reduce the loudpeaker requirement to le than octave beyond the croover frequency. owever, a paive implementation i practical only for the all-pa filter, ince component count would be too large with the three-way ytem. A fourth-order tranfer function i defined from N with N 4: a a a a a a.4.. Symmetrical three way A ymmetrical three-way croover network can be build uing the middle term for the midrange loudpeaker. Coefficient a mut be removed from the nominator a a a a a a a a a a a a P BP P An active implementation could ue: P P BP The low-pa and high-pa channel include two term each o they cannot be implemented uing tandard low-pa and high-pa filter but the circuitry i not too complex to be implemented uing active filter. When build, the midrange channel i derived by ubtracting the channel from the input ignal. Aume that the coefficient are a, a and a, to implify the analyi. The following amplitude and phae can then be found at the croover frequency ( i): i i i i i i i i i i P BP P So, the low-pa and high-pa channel are 486 out of phae (correpond to 6 ) and add with ome lo and the midrange channel add the required ignal. The channel are all at fairly high level around croover o thi i a deign, which i baed upon ubtraction of large figure it hould be avoided.

35 oudpeaker croover network Figure 4 Amplitude and phae with a, a, a..4.. Steep cut-off two way The term with a, a and a are removed from the nominator reulting in a two-way croover with maximum cut-off lope within both channel. The filter doe not atify the requirement of unity tranfer function. P4 P4 a a a 4 a a a etermination of the coefficient aume modelling by a combination of two econdorder Butterworth filter in cacade (the inkwitz-iley method). The nominator polynomial i not important for thi evaluation, only the denominator i conidered. P4 P P N c c By comparion, the coefficient are found to: a c N c N N ( c ) c a c.8 a 4. c.8, where c A paive implementation i at the limit of what can (or hould) be done but the network i traight forward from a theoretical point of view. It i a requirement that the loudpeaker are impedance compenated. Ørted TU Acoutical Technology 5

36 oudpeaker croover network B CT B CT CB CB Ba B T T Treble T Figure 5 Paive croover network. The reult i hown in Figure 6. The phae difference between the channel are 6 at croover o the ignal are added without lo. Figure 6 Amplitude and phae repone with a a.8, a 4.. Figure 7 Group delay with a a.8, a 4.. Ørted TU Acoutical Technology 6

37 oudpeaker croover network.5. Paive network A paive network i bet uited for low order croover network and can be build a hown in Figure 8, where,, etc. are impedance, which may conit of reitor, inductor and capacitor or even combination hereof. The number of branche i defined by the filter order; a firt order croover network would conit of only. Figure 8 Conventional ladder-network for a paive croover. The impedance can be any of reitor, inductor or capacitor. The tranfer function of the filter can be derived from inpection of the circuitry. The below collection of croover network tranfer function i limited to third order ince higher order network become more involved and are of little practical ue. igher order filter hould preferably be build uing active circuitry..5.. Firt order Thi conit only of o not ued and i a hort circuit. The filter i a voltage divider between the erie element and the loudpeaker, repreented by the C voice coil reitance. The tranfer function for the firt order network i: To introduce real component, the impedance mut be ubtituted by an inductor or a capacitor. The impedance of the inductor and capacitor i defined a: C C Uing an inductor for the reult i a low-pa filter: P, where ω Uing a capacitor for the reult i a high-pa filter: C ω P, where C C C Ørted TU Acoutical Technology 7

38 oudpeaker croover network Ørted TU Acoutical Technology Second order The croover network include, which i hunted acro the loudpeaker (ince i a hort circuit), o the tranfer function can be derived from by ubtituting by a parallel combination of and : Uing an inductor for and a capacitor for the reult i a low-pa filter: C C C P After reduction: C Q a C where a P, ω Uing a capacitor for and an inductor for the reult i a high-pa filter: P C C After reduction: C Q a C where a P, ω.5.. Third order erivation tart from the loudpeaker where form a voltage divider with. The two firt component alo form a voltage divider between and, where i in parallel with. ( ) ( ) After reduction::

39 oudpeaker croover network Uing an inductor for and and a capacitor for the reult i a low-pa filter: P C After reduction: C C C ( ) C P ω a C a a C a C C C Uing a capacitor for and and an inductor for the reult i a high-pa filter: C C C C C C C After reduction: P ω a C a a C a C C.5.4. Fourth order Filter realiation will ue the ladder-network hown in Figure 9, which i a general filter uing impedance. G 4 Figure 9 Generic ladder-filter for realiation of the croover network. The impedance repreent any combination of capacitor ( /C), inductor ( ) or reitor ( ). The load reitor repreent the loudpeaker. The voltage at the node are, according to Kirchhoff law, which tate that the um of current to a node mut equal zero (with poitive direction defined away from the node): G 4 Ørted TU Acoutical Technology 9

40 oudpeaker croover network Ørted TU Acoutical Technology 4 The term are rearranged: G 4 4 limination of reult in: 4 G And: 4 G And: G 4 The tranfer function become: 4 G.5.5. oudpeaker impedance All paive croover network are enitive to the load impedance and they are mot often deigned for contant and reitive loading; but the impedance of a loudpeaker i neither contant nor reitive a can be een from Figure 4. The impedance i real only at very low frequencie and change from capacitive to inductive within the pa band.

41 oudpeaker croover network Figure 4 lectrical impedance of loudpeaker with reonance frequency normalied to unity, with voice coil cut-off frequency et to time the reonance frequency and with mechanical quality factor Q M 5. A paive croover network will not operate a intended if loaded by thi impedance o the preumption of the filter will be validated and the reult can be rather dramatic. C C Figure 4 Compenation network for loudpeaker impedance correction. Conider a treble loudpeaker with kz reonance frequency, which i to be activated above 4 kz by a erie capacitor. The required attenuation at the croover frequency i ( kz)/(4 kz).5, o the capacitor mut have an impedance of 4 time the nominal impedance at thi frequency. But the impedance of the loudpeaker may have increaed to 4 time the nominal value, thu compenating for the rie in impedance; the effective croover frequency become the reonance frequency of the loudpeaker. It i poible to compenate, at leat partially, for the variation in loudpeaker impedance, uing network hown Figure 4, thu introducing additional component increaing complexity and cot. The equation for the correction network are hown below []. The required parameter are the electrical and mechanical quality factor (Q C and Q MC ) and the reonance frequency (f C ) of the loudpeaker in the cloed cabinet, the C reitance of the voice coil ( ), and the inductance of the voice coil ( ). Q Q QC πf C C C MC C πf Q C C π Ørted TU Acoutical Technology 4

42 oudpeaker croover network Midrange loudpeaker: Q C.4, Q MC, f S 8 z, 6 Ω.and. m. Compenation of the reonance frequency: 6.8 Ω, 4.8 m and C 8 µf. Compenation of the voice coil inductance: 6 Ω and.44 µf. It i not required compenating for the reonance frequency of a ba loudpeaker, ince thi i within the pa band of the croover network and imilar argument are valid for the voice coil inductance of the treble loudpeaker. A midrange loudpeaker may require compenation of both parameter. An active croover network olve the problem of interaction between the croover network and the loudpeaker. It offer better control of filter parameter, ince inductor can be avoided, and it i inenitive to the temperature dependency of the voice coil C impedance. Problem, which mut be addreed by the deigner of a paive croover network, in addition to the more obviou problem of electing the croover frequencie, deciding a network topology and how to contruct the loudpeaker ytem from available component..6. Active network Active network are build from filter block, uch a the two block hown in Figure 4. Amplifier are ued a buffer avoiding interaction between the filter ection. It hould be mentioned that the layout poibilitie are large and the below example repreent but a few of the poible implementation. Firt-order filter block Second-order filter block Power amplifier and loudpeaker Buffer 4 Buffer A 5 B 6 Figure 4 Active filter coniting of a firt-order filter block and a econd-order filter block. The amplifier iolate the ection thu implifying the deign. Active filter are deigned from reitor, capacitor and amplifier circuit. The inductor i not required and ha been omitted ince it i difficult to build good inductor; they are plagued by their erie reitance and parallel capacitance and the component i prone to pick-up of hum from magnetic field. The reitor and capacitor are almot ideal component with few paraitic component. The miing inertia, offered by the inductor in the paive croover network, and required by circuit with complex pole, i upplied by the amplifier. In thi example i the amplifier arranged a a voltage-follower, which mean that it monitor the voltage at the input without loading the circuitry and output the voltage. The gain factor (amplification factor) i unity. Ørted TU Acoutical Technology 4

43 oudpeaker croover network Ørted TU Acoutical Technology Firt order The filter i actually a paive filter ince one of the component are a reitor and the other a capacitor. The network i energied from node with output at node. C C C C P C P C / / ω.6.. Second order The tranfer function i bet derived from the Kirchhoff law of node, which tate that the um of current into a node mut equal zero. There are two internal node, A and B, but node B i identical to the output node due to the buffer. The network i energied from node with output at node. The equation are: A A A A After reduction: Uing reitor for 4 and 6 and capacitor for and 5 reult in a high-pa filter:, C C C C Q a C C where a P ω Uing reitor for and 5 and capacitor for 4 and 6 reult in a low-pa filter:, C C Q a C C where a P ω.6.. igher order Filter of higher order can be build from firt and econd order filter block by cacading the filter. A eventh order two-way croover network will be ued a an example: 7 7 a a a a a a P P

44 oudpeaker croover network All block hare the ame cut-off frequency, given by ω, with the Butterworth deign but the coefficient a, a and a, are different. The coefficient determine the root of the denominator polynomial. The root of the firt order polynomial i, and the root of the econd order polynomial are determined from: The root are: a a ± α ± iβ a 4 a ± i 4 a It follow that both the real part and imaginary part of the root (α and β repectively) are defined by the coefficient a. ence, the value i given by: a α For the eventh order Butterworth polynomial are the root defined a hown below in Table, which alo how the calculated value of the coefficient a well a the quality factor Q /a: Table Coefficient for a eventh order croover network. The real value of the root determine the coefficient and thu the quality factor. Section oot Coefficient a Quality factor Q (-) - - (-.5 ±i.9749) (-.65 ±i.788) (-.9 ±i.49) The coefficient could a well be determined from the imaginary root value, which give the ame reult. Take for intance the lat root. With a.8 i imaginary value: ± β Thi i indeed the pecified value. 4 a ± 4 (.8) ± Special Several of the croover network ued an algorithm uch a the following for derivation of the low-pa channel from a teep high-pa channel. P P Thi can relative eaily be implemented a hown below where the input ignal i highpa filtered and routed through a power amplifier to the treble loudpeaker a well a to a ubtraction circuitry for contruction of the low-pa channel. The high-pa filter i realied a a conventional third-order filter and a ubtraction network (an operational amplifier) i ued to generate the low-pa channel. Ørted TU Acoutical Technology 44

45 oudpeaker croover network Ørted TU Acoutical Technology 45 Power amplifier igh-pa filter Ba Power amplifier Σ Treble - Input a a a a a a a a Figure 4 Active croover network.

46 oudpeaker croover network. Model A loudpeaker i not jut a linear tranducer that output ound a a true replica of the input ignal; the loudpeaker, the baffle and the litening room introduce limitation, everely affecting the ignal quality. The loudpeaker bandwidth i limited and it i not mall compared to wavelength at high frequencie, which affect the radiation angle concentrating the ound at the front of the loudpeaker. The front baffle introduce an impedance dicontinuity at the edge, which caue reflection known a diffraction, that interfere with the direct ignal and reult in the well-known 6 db lo of ba a well a ripple at higher frequencie. itening off-axi introduce time difference for ignal from the different loudpeaker, which create ripple around the croover frequency, and reflection from large urface further affect the low frequency reproduction. The model derived in thi ection i a tranfer function for analytical tudie uing MATAB. The model apply olely to the electro-dynamic loudpeaker uing a moving-coil for the energy tranfer; o electrotatic, piezoelectric and ribbon-type loudpeaker are not referenced. At firt, a model for the loudpeaker i introduced and thi i followed by model for the urrounding; i.e. diffraction due to the baffle, the litening angle due to the ditance between the loudpeaker and the reflection from the boundarie of the litening room. A model i alo introduced for the calculation of group delay baed upon the reultant tranfer function... lectro-acoutical model The tranfer function model for the loudpeaker conit of everal term each decribing a pecific part of the loudpeaker. The reult i the ound preure p at ditance r with an number of parameter decribing the loudpeaker. r r ( ikr) K G F p VC exp The excitation voltage G from the power amplifier i converted to a ound preure by the contant K, which aemble the electrical, mechanical and acoutical parameter and repreent the ound preure within the middle of the pa-band at a reference ditance r F, but it doe not include parameter uch a frequency and angle. The ueful frequency range i pecified by VC for the voice coil high-frequency cut-off and by for the low frequency cut-off. The invere-ditance law i pecified by r F /r, and the exponential function. The model can eaily be enhanced by introducing more function dealing with pecific area. erivation tart by analying the mechanical contruction of the loudpeaker and the following analyi will be baed upon the introduction preented by each and with reference to Beranek. The loudpeaker model applie to ba, midrange and treble unit and will be limited to unit with the rear ide radiating into a cloed cavity thu realiing a ingle model common to all the loudpeaker unit. Ørted TU Acoutical Technology 46

47 oudpeaker croover network Voice coil current I C Generator voltage G lectrical ytem Voltage G, M Current I C Mechanical ytem Force F M, F A Velocity U Acoutical ytem Preure p Volume velocity S U Volume adiation velocity reitance S U Front AF ear A Sound preure p Figure 44 The model i baed upon the model of a loudpeaker and operate with three different ytem analogie: electrical, mechanical and acoutical. The initial part of the analyi concentrate on deriving a model for the diaphragm velocity a a function of generator voltage, which introduce the loudpeaker frequency dependency. The diaphragm velocity i converted into ound preure, which introduce the invere-ditance law. The remaining analyi deal with the conequence of diaphragm diameter, monitoring angle and the baffle ize thu introducing directivity and diffraction. The conequence of reflection from the litening room will alo be conidered. It i common to divide the loudpeaker unit into three different ytem analogie a hown in Figure 44. The electrical ytem i reponible for the voice coil current, the mechanical ytem tranlate thi into diaphragm velocity and the acoutical ytem account for ound preure, air loading and the effect of the cloed cabinet.... The loudpeaker unit A layout of the electro-dynamic loudpeaker i ketched in Figure 45, which alo diplay the main difference between the ba/midrange and treble unit. The diaphragm of the treble unit i reduced to a dome to cut the ma and the ize of the loudpeaker thu improving reproduction of the high-frequency range. The diaphragm i the moving urface of the loudpeaker which radiate ound. The preferred material i paper, due to it low weight and high internal damping, although platic and metal are alo ued. The diaphragm i cone or dome haped to improve rigidity although loudpeaker with flat diaphragm exit. A upenion ytem i ued to keep the diaphragm at the correct poition with the voice coil within the magnet gap and retrict the diaphragm movement to the axial direction only. The effective diameter of the diaphragm () include ome of the outer upenion, which i moving a well, o the effective cro-ectional area become: S a π π 4 The area i cm for an 8 inch loudpeaker and 5 cm for inch diameter. For treble loudpeaker i the diaphragm a dome and the equation i give a rough etimate when the dome i approximately one-quarter of a phere. Ørted TU Acoutical Technology 47

48 oudpeaker croover network Mounting bae Magnet Air gap Supenion a Mounting bae Supenion Voice coil ut cover iaphragm (cone) iaphragm (dome) Maximum linear diplacement A ide of the diaphragm FONT ide of the diaphragm Figure 45 Model of ba and midrange unit (left) and treble unit (right). The main difference between unit i the ma of the moving ytem and the effective croectional area of the diaphragm. The loudpeaker will be aumed ealed to interrupt the radiation from the rear of the diaphragm thu reducing the complexity of the model. The intention i that the ame model hould apply to all loudpeaker unit with adjutment of parameter value. Treble unit are almot alway ealed and midrange unit mot often include a can at the rear ide. The ba loudpeaker are never ealed and mut be placed within a cloed cabinet. A will be hown later in thi ection, the loudpeaker i a tranducer where the tranfer function i a band-pa filter with a frequency range determined by the moving ytem and the voice coil. The ound preure i downward limited by the reonance frequency below which any cloed-box loudpeaker ytem will drop off at db/octave and the ound preure i upward limited by the electrical low-pa filter of the voice coil reitance and inductance above which the repone drop off by 6 db/octave. The model aume that the loudpeaker diaphragm i vibrating a a rigid piton. Thi i true for low frequencie where the diaphragm circumference πa i hort compared to wavelength λ. Thi i mot often expreed a ka <, where frequency i repreented by the angular wave number k: ω πf π c ka < k f < c c λ πa A loudpeaker with diameter a.5 m become directive above approximately. kz. A loudpeaker can be aumed equivalent to a monopole ound ource at low frequencie (ka < ) where radiation i equally well in all direction. The loudpeaker become directive at high frequencie (ka > ) where the output become concentrated on the loudpeaker axi and the off-axi output i limited. At higher frequencie (ka > ) may the diaphragm break up and vibrate in ection with different phae, which affect both amplitude and directivity. Ørted TU Acoutical Technology 48

49 oudpeaker croover network... lectrical circuit lectrical variable are the voltage difference and current I and the electrical impedance i defined by: V unit : Ω I A Input to the loudpeaker i an applied voltage G from an external voltage generator with an internal erie impedance G, ee Figure 46. The reulting voice coil current I C generate a voltage drop acro the generator impedance of: G I C The voice coil current flow through the voice coil reitance and inductance. The voice coil i hown in parallel with a reitor to model the effect of eddy current loe in the magnetic circuit. The voltage acro the voice coil i given by: When the voice coil and diaphragm i moving with velocity U, a voltage i induced into the voice coil wire: B M U The product of the magnet flux denity B and the effective length of the voice coil wire i the force factor, which i pecified in the loudpeaker data heet. We now know the individual term of the electrical ytem. Generator impedance G Voice coil current I C Voice coil reitance I C Voice coil inductance Generator voltage G ddy current Voice coil voltage M xternal Internal Figure 46 lectrical model of loudpeaker input with the generator G repreenting the power amplifier and with the voice coil C reitance and the inductance. The feedback from the moving ytem i repreented by the econd voltage ource M. The relation between generator voltage G, voice coil current I C and diaphragm velocity U can now be derived by Kirchhoff law, which tate that the um of all voltage within a cloed mak mut equal zero. It follow that: G G IC The generator impedance i often ignored ince the output impedance approximate zero in mot ituation. Thi aumption will be ued to implify the expreion, but the G can be re-introduced at any time by ubtituting with G. M Ørted TU Acoutical Technology 49

50 oudpeaker croover network ence, the voice coil current become: I C G M G The equation include a filter with a cut-off frequency (pole frequency f C ) due to the voice coil erie reitance and inductance and a correction due to the eddy current loe repreented by (null frequency f ). The frequencie are: M f f C ω C π π ω π π π f C -6 db/octave f For 5 Ω and m i f C 8 z (ignoring ) o the higher frequencie are attenuated by 6 db/octave. For Ω i f 6 kz above which the attenuation ceae. The voice coil current i repreented by: I C VC G The tranfer function VC due to the voice coil i defined by: VC ω ω ωc ω C M For low frequencie i VC, which indicate that the voice coil current I C i proportional to the difference between the excitation voltage G and the velocityinduced voltage M and that the correlation contant i the C reitance of the voice coil. For high frequencie are the voice coil current reduced ince the impedance of the inductance i becoming the dominating term.... Mechanical circuit Mechanical variable are force F and velocity U, which are analogue to voltage and current in electrical ytem although the electrical current flow through erieconnected component, wherea the velocity in mechanical ytem i common to parallel-connected component. The mechanical impedance i defined by: F N M unit : U m kg C Ørted TU Acoutical Technology 5

51 oudpeaker croover network The motor of the loudpeaker i the voice coil, which i located within the trong magnetic flux of the air gap. The electrical current I C within the voice coil reult in a mechanical force working on the voice coil: F B M I C A typical value of B i N/A o a voice coil current of one ampere reult in a force on the voice coil of N, which i approximating the weight of a kg plumb. A preure difference between the front and rear ide of the diaphragm reult in a force working on the diaphragm, which i the preure difference multiplied by the area: A ( pf p ) S p F S For an 8 inch loudpeaker with a diaphragm area S. m, and a preure difference of p Pa, correponding to 94 db ound preure level, the force from the acoutical ytem become f A. N, which i mall compared to the mechanical force involved o the reaction from the acoutic ytem can be ignored in ome application. The reultant force working upon the mechanical ytem become: F F The direction of the force ha been elected o F M > drive the diaphragm forward while F A > drive it backward. Thi i an arbitrary choice, but it model the actual behaviour of the loudpeaker where the mechanical force i oppoed by the force from the acoutical ytem. lectrical force f M Moving ma M M M Acoutical force f A F Supenion compliance C MS A iaphragm velocity u Supenion loe MS Figure 47 Mechanical model of loudpeaker moving ytem (mobility analogy). The ma of voice coil and diaphragm are aembled into M M, the upenion ytem i repreented by the pring compliance C MS and the friction loe by MS. Typical value for an 8 inch loudpeaker are: M M 8 - kg, C MS 5-6 m/n and MS kg/. The mechanical ytem conit of the ma M M of voice coil and diaphragm, and of the compliance C MS and friction loe MS of the upenion ytem. The diaphragm velocity U i decribed a the olution to the following differential equation: M M du dt MSu C MS u dt F M F The firt term i due to the law of motion, F M M M a, where a du /dt i the acceleration of the diaphragm. The econd term i the relation between force and friction loe, F MS u. The third term i due to compliance, F C x /C MS, where x i the diaphragm diplacement and i related to velocity by u dx /dt. A Ørted TU Acoutical Technology 5

52 oudpeaker croover network The mechanical ytem i completely decribed by the reonance frequency f S and the quality factor Q S. f S Q S π In the frequency domain i the equation eay to olve for diaphragm velocity: Thi can be written: M U M F M M F A MS MS M C where MS M M C C M MS U M MS F M The correponding circuit model i hown in Figure 48. iaphragm velocity U Moving ma M M Compliance upenion C MS Supencion loe MS M M F A MS C MS lectrical force F M Acoutical force F A Figure 48 Impedance analogy of the mechanical ytem. The ma and compliance cancel at the reonance frequency where the diaphragm velocity i maximum. The diaphragm velocity i maximum around the reonance frequency where M MS and i in phae with the excitation. Below the reonance frequency will the diaphragm velocity increae with increaing frequency and the phae approache 9, while the diaphragm velocity decreae with frequency above the reonance frequency where the phae approache 9, compared to the excitation...4. Acoutical circuit Acoutical variable are the preure difference p and volume velocity SU, which are analogue to voltage and current in electrical ytem. The volume velocity i in thi tudy repreented by the diaphragm cro-ectional area S multiplied by the velocity, hence the variable SU. The acoutical impedance i defined by: p N kg A unit : SU m m Two different load impedance will be analyed below; the loading at the front ide of the diaphragm, which i aumed radiating into free pace, and the loading at the rear ide, which i radiating into a cloed cabinet. The front ide of the diaphragm i loaded by air and the impedance i defined in [] for a circular piton at the end of an infinitely long tube. The impedance conit of a ma term, which dominate for ka <, and a frequency-dependent reitive term, which Ørted TU Acoutical Technology 5

53 oudpeaker croover network dominate at higher frequencie and approximate a contant for ka >. The reulting impedance i a parallel combination of: M AF ρ ρc.95 and AF. 8 a a xample are M AF. kg/m 4 and AF kg/m 4 for a. m (8 inch loudpeaker). The impedance are tranformed to the mechanical circuit by multiplication with S. For S πa - m the value become: M MF. g and MF kg/. The ma term repreent a volume of uncompreed air, which i moving with the diaphragm velocity. The reitance term repreent the energy lot into the air. Since the component are in parallel the total impedance become: AF M M AF AF AF AF M AF ω AF where The cut-off frequency f AF i.4 kz for the 8 inch loudpeaker. f AF ωaf π πm A fair repreentation i a ma M AF at low frequencie (F) and a reitor AF at high frequencie (F), which will be repreented a follow (ka limit from []): AF ( F ) ( F ) M for ka <.5 and for ka > 5 AF The rear ide of the loudpeaker i loaded by the cloed cabinet, which i repreented by an acoutical impedance with C AB for the compliance of the confirmed air, M AB for the air load on the rear ide of the diaphragm and AB for the loe within the box. The acoutical lo within the cloed box AB cannot eaily be calculated and mut be etimated by other mean; but luckily, the value i of minor importance, at leat for thi tudy, and can afely be ignored. The compliance and ma load can be calculated by the following approximation, which aume that the box i mall and without any damping material. C AB VAB ρ c and M AB AF AF ρ.65 πa A box i mall when the larget dimenion i le than λ/, which correpond to the following deign requirement where B repreent the larget box dimenion: c f > The equivalent circuit for the loading of the loudpeaker i hown in Figure 49, with the rear of the loudpeaker radiating into a cloed box and the front into free pace. B AF AF Ørted TU Acoutical Technology 5

54 oudpeaker croover network Box compliance C AB Acoutical Moving air loe ma AB M AB Volume velocity S U Air load ma M A p Sound preure difference Air load reitance A Figure 49 oad impedance for the loudpeaker. ence, the ound preure difference between the loudpeaker front and rear ide: p M C AB ( F ) ( F ) AF AF M AB AB SU ( AF A ) SU adiation reitance AF i a function on loudpeaker mounting. The loudpeaker may be approximated by a monopole ound ource at low frequencie but the loudpeaker baffle come into play at higher frequencie and the model i then approximately a monopole cloe to an infinite wall, which double the ound preure. The tranition between the two model i unfortunately not traightforward. The model hould alo include the interaction between two ource operating at the ame frequency (i.e. at cro-over) and the reflection from cabinet boundary (diffraction). In order to eparate the different model, the loudpeaker model will load the front of the loudpeaker by the radiation impedance of a plane piton in an infinite tube. Thi model the monopole ound ource and the other factor can then be added to the model without a need of changing the baic model...5. iaphragm velocity The firt tep i to derive a relation between the diaphragm velocity U and generator voltage G and the final tep i to determine the ound preure. Two expreion are at hand for the voice coil current I C and uing them to eliminate the voice coil current introduce G, U and the force F M from the voice coil. I C VC G BU FM B F M VC B G ( B) VC repreent the voice coil high-frequency attenuation, which i unity for frequencie below approximately 5 z. Another expreion for F M i available introducing the mechanical impedance M and the force from the acoutical ytem F A, which again introduce the acoutical impedance AF and A for the loading on the loudpeaker front and rear ide. F M M U F A F M M U ( AF A ) SU limination of F M reult in the following equation for etimation of diaphragm velocity U veru generator voltage G. VC B G ( B) U U M ( AF A ) SU U Ørted TU Acoutical Technology 54

55 oudpeaker croover network The expreion i olved for diaphragm velocity U a a function of G : B U VC ( B) VC M ( AF A ) S The firt term i conit of mechanical impedance with the voice coil VC tranformed to the mechanical ytem by (B), and the acoutical impedance tranformed by S. The (B) VC / term repreent the damping from the electrical ytem. The term i active around reonance where the denominator conit of (B) / plu the frequencyindependent term from M, AF and A (the reactive term cancel at reonance). The (B) / term i gradually removed at high frequencie due to VC, but the denominator i dominated by the ma-term of M above reonance, o VC can be ignored without conequence at higher frequencie. Thi can be een from the following inequity, which tate the requirement for the ma term to dominate over the (B) / term: ( B) ( B) < M M ωm M f > π M For B N/A, 5 Ω and M M.5 kg i f >6 z. The VC term i unity for frequencie below approximately 5 z, o the correction due to VC can be ignored. Uing thi implification and introducing the definition for the mechanical impedance M and the radiation reitance AF and A the equation become. U ( B) ( F ) ( F ) M M MS C MS VC M B AF G AF M M AB AB G C A tated before, the acoutical impedance for the front of the diaphragm conit of two contribution, the ma M AF, which i effective at low frequency (ka <.5), and the reitance AF, which i effective at high frequency (ka > 5). Note that the two term are not active at the ame time, hence the notation with uper fixe F and F. A will be een in the following, the reitance term AF i of no importance for the definition of the reonance frequency, ince the reonance frequency thi i a lowfrequency parameter, and it will be completely maked by the ma of the moving ytem at high frequencie, hence AF will never contribute to the tranfer function and will be ignored. The above implification do not limit the ueful frequency range of the model; we are imply deleting term, which will never contribute to the reult. The model i thu valid from the extreme low-frequency range and up to the limit where the loudpeaker diaphragm break up, i.e. about around ka. AB S Ørted TU Acoutical Technology 55

56 oudpeaker croover network The following definition will be introduced to eae the analyi: M MS, which i the moving ma of the total ytem, MT, which i the mechanical reitance of the total and C MT, which i the mechanical compliance of the total ytem: M MS MT C MT M M ( B) C MS ( M M ) MS AF S C AB AB AB S C S MT C S C MS MS CAB C ence, the following equation for the diaphragm velocity: MT AB C C B U VC M MS MT C MS G MS C S C S The equation i multiplied and divided by MT and the nominator and denominator are multiplied by C MT thu reulting in a econd-order denominator polynomial in. C B MT MT U VC M MSCMT MSCMT MT The reonance frequency f S i identified a the frequency where M MS C MT : f S ω π S π The econd-order term i normalied to (/ω S ) and the firt-order term MS C MT i normalied a follow: M MS C MT AB G AB MT C MT ωs ω S MT C MT ω S M MT C MS MT C MT ω S MT C M MT MS The contant factor to /ω S i dimenion-le and i identified a the total quality factor for the loudpeaker in a cloed cabinet: Q TS MT The total quality factor i enitive to both the electrical, mechanical and acoutical ytem but damping from the electrical ytem i the dominating factor. Q TS.5 correpond to a critically damped ytem, which hould theoretically reproduce tranient without ocillation. The level i 6 db down at the reonance frequency, which can be compenated electronically by a ba boot from the amplifier. Q TS.7 correpond to a Butterworth filter with db at the reonance frequency and ome ocillation with tranient input. It i often called the maximally flat alignment. M C MS MT Ørted TU Acoutical Technology 56

57 oudpeaker croover network Q TS. correpond to a db Chebychev filter with db at the reonance frequency, a peaking of db above the reonance frequency and ocillation with tranient. It i common to divide the quality factor into electrical and mechanical quality factor by the ue of the definition for MT and ignoring the acoutical reitance, which i mall compared to the two other. Q S M MS Q MS TS ( B) CMS MS CMS QS QMS The quality factor i proportional to. Thi reflect the recommendation, found in many loudpeaker book and magazine, that a ba loudpeaker hould not be driven from a high-impedance ource. Thi would otherwie increae the effective value of (which i in erie with G ) and reult in a booming ba. The tranfer function for diaphragm velocity U a a function of generator voltage G can then be expreed a follow: TS S U VC MT ωs Q ω Q TS ω S The expreion for the diaphragm velocity i a band-pa filter centred at the reonance frequency and with ymmetrical kirt. The diaphragm velocity i maximum at the reonance frequency and le for all other frequencie...6. Sound preure erivation of the ound preure from the loudpeaker ue the equation for the ound preure at ditance r from a monopole ound ource in free pace. The ound preure i pecified with a given volume velocity S U and i []: p iωρ 4πr M MS ( r) exp( ikr ) S U The exponential i a complex unit vector which define the phae due to the delay of the ound wave travelling the ditance r from the ource to the monitoring point where k i the angular wave number (k ω/c). The ound preure become: iωρ Q ω B p 4πr MT ωs QTS ωs TS S ( r) exp( ikr) S VC G Q B Q G S Q MS Ørted TU Acoutical Technology 57

58 oudpeaker croover network Which can be rearranged into five term: two tranfer function, one angular tranfer function, a contant expreion and the excitation voltage: iω ω ρ ω S B p 4πrQTS MT ωs QTS ωs S S ( r) VC exp( ikr) G The reonance frequency ha been included into the contant expreion in order to implify the expreion. Uing the relation iω and introducing a reference ditance r F, the reultant tranfer function for the ound preure p(r) at ditance r given by the excitation voltage G : r p r F ( r) VC exp ( ikr) KG The tranfer function due to the mechanical ytem i: ωs ωs QTS The contant K aemble the caling factor: ρ f SSB K r Q F TS MT ω xpreing MT by the definition of the total quality factor and ubtituting C MT by the definition of the reonance frequency, the mechanical loe can be expreed a: MT The caling factor i thu: Q TS M C MS MT Q TS F S S M ω M ρsb K 4πr M MS MS MS M MSωS Q The loudpeaker i ma-controlled above the reonance frequency, i.e. the diaphragm movement i controlled almot entirely by the force from the voice coil and the ma of the moving ytem M MS. Thi repreent the normal ue of the loudpeaker. The caling factor i valid for all frequencie but aume a diaphragm vibrating a a rigid piton. The effective frequency limit i thu approximately ka < : f MAX c π a For an 8 inch loudpeaker with a. m the upper limit i approximately.6 kz. TS Ørted TU Acoutical Technology 58

59 oudpeaker croover network.. oudpeaker pa band oudpeaker are band-pa filter with a tranfer function given by VC from the previou ection. The amplitude repone i hown in Figure 5 for a theoretical loudpeaker with the frequency axi normalied to the reonance frequency. Figure 5 epone of a loudpeaker with ytem reonance frequency normalied to unity, total quality factor Q TC.7 and the voice-coil frequency at time the ytem reonance. The ueful pa band i the frequency range above the mechanical reonance frequency (normalied to unity) and below the cut-off due to the voice coil inductance (arbitrarily et to time the reonance frequency). Frequencie below the mechanical reonance frequency are attenuated db/octave and high frequencie are attenuated 6 db/octave due to the voice coil inductance. The actual performance may depart from the typical view hown above due to the contruction affecting the voice coil impedance, the loudpeaker directivity and diaphragm break-up.... Sound preure level The ound preure level i calculated from: ( r) log p p ( r) F db The reference ound preure i p F -6 Pa and correpond to db of ound preure level, i.e. the threhold of hearing around kz. The reference ound preure level F, at ditance r F (typically m) and excitation voltage F (typically.8 V for W of power diipated into 8 Ω) and without the influence from a loudpeaker baffle (radiating into 4π), i: F 4πr ρ S p B M ( 4 π ) log F db F F MS Ørted TU Acoutical Technology 59

60 oudpeaker croover network The reference ound preure level i ued to characterie the loudpeaker enitivity and i mot often meaured with radiation into π. In order to compare the calculation to the meaurement the formula can be changed to radiation into π: F πr ρ S p B M ( π ) log F db F Meaurement on the loudpeaker typically ue a rigid plane that form one of the wall of the anechoic chamber. Thi i a practical implementation of an infinite baffle. xample. Peerle 5SW: S 5 - m, B.6 N/A, 5.5 Ω, M MS 84 - kg, and F.8 V. The enitivity at r F m become 9.6 db re. µpa. The pecification i F 89. db, which i. db below the calculated. xample. Peerle 5W: S - m, B 9.6 N/A, 6. Ω, M MS 6 - kg, and F.8 V. The enitivity at r F m become 9. db re. µpa. The pecification i F 9 db, which i. db below the calculated.... iaphragm excurion oudpeaker deign require obervation of the diaphragm excurion in order to deign a loudpeaker that can withtand the intended ue. An important parameter i the diaphragm excurion, ince the ytem may depart from the aumption of linearity at high level and low frequencie. The analyi tart from the previouly derived expreion for diaphragm velocity: TS F TS S U VC MT ωs Q ω Q ω The econd-order function i a band-pa filter with a maximum at the ytem reonance frequency f S, o the diaphragm velocity i proportional to frequency below reonance and inverely proportional to frequency above reonance. The loudpeaker i aumed ued above the reonance frequency and ince the following analyi will deal with low frequencie can the firt term be aumed unity. A good approximation for the diaphragm velocity above the reonance frequency and below the voice coil cut-off frequency i obtained by oberving that the (/ω) term dominate and the VC i unity. The diaphragm velocity can thu be decribed by: U o U ω > ω S ω< ω Q C TS S MS MT ωs B B The equation can be rewritten with the aid of the definition of Q TS : ω B B U o S G QTS MT M MS iaphragm excurion can be calculated from: x Frequency tranformation ( t) U ( t) dt x ( ) G G G ( ) U iω Ørted TU Acoutical Technology 6

61 oudpeaker croover network ence, and ignoring the minu ign from (iω) : B x ω M The lowet allowable frequency (above reonance frequency) for a maximum excurion x and generator voltage G i: MS G f MIN π B G x M MS xample. For B N/A, G 4 V ( G MS 8. V for W into 8 Ω), x 5 mm, M MS g and 6 Ω the limit i 6 z. With G 4 V (for W into 8 Ω) the ame loudpeaker can operate down to 5 z. For inuoidal excitation at x excurion the ocillation i decribed a: ( t) e{ x exp( i t) } x ω The maximum diaphragm velocity i found a the maximum of the derivative of the ocillation: u MAX ( t) dx e exp dt MAX The maximum free field ound preure become: p iωρ 4πr { xiω ( iωt) } ωx MAX f S x r πρ ( r) exp( ikr) S ωx exp( ikr) The maximum ound preure level at frequency f and at the reference ditance r F and with an excurion of x i: MAX ( 4π ) log db i πρ f S x rf pf xample. For an 8 inch loudpeaker with S - m and x 5 mm excurion will the maximum ound preure level at r F m and f 5 z be MAX 9 db. A inch loudpeaker with S 6 - m and ame linear excurion would produce MAX 97 db.... SPIC imulation model Baed upon the previou ection, the loudpeaker imulation model become a hown in Figure 5. lectrical ytem Mechanical ytem Acoutical ytem p G MMS CMS MS Q G IC M FM U FA PB AB AF PF Figure 5 oudpeaker model with input from the generator ( G ) at the left ide and output at the front ide of the diaphragm (p F ) at the right ide. Ørted TU Acoutical Technology 6

62 oudpeaker croover network.. irectivity A loudpeaker doe not ditribute the radiation evenly acro pace although it i often aumed to do o at low frequencie in order to implify thing. A diaphragm diameter or frequency increae, the loudpeaker concentrate the radiation on-axi. The ound preure at ditance r for a circular piton with radiu a and volume velocity Q S U in an infinite baffle i []: p ( r θ ) ( kain( θ )) in( θ ) iωρq J r ka exp π, ( ikr) The expreion within the quare bracket repreent the directivity. The on-axi repone i obtained for kain(θ) mall where the Beel function can be approximated: kain J ( kain( θ )) ( θ ) Uing thi approximation the on-axi ound preure become: p iωρq πr ( r, θ ) exp( ikr) Thi how that the far-field on-axi preure radiated by a circular dik in an infinite baffle i identical to the preure radiated by a point ource againt a rigid urface. The concluion i that the directivity of a loudpeaker in a cabinet can be modelled by: J ( θ ) ka ( kain( θ )) in( θ ) The directivity i plotted in Figure 5. ow frequencie (ka < ) are virtually unaffected by the obervation angle (le than db attenuation) while high frequencie (ka > ) are attenuated for the off-axi radiation. Figure 5 irectivity for a loudpeaker in an infinite baffle at four different value of ka, where a i diaphragm radiu. The directivity i of minor importance for ka > ince the loudpeaker may uffer from diaphragm break-up, which dratically change both amplitude and directivity, and thi behaviour i hard to model. Ørted TU Acoutical Technology 6

63 oudpeaker croover network.4. iffraction oudpeaker unit are in thi report aumed mounted within one of the wall of a cloed cabinet o the proximity of the loudpeaker i a rigid urface. The cabinet dimenion can be conidered mall only at low frequencie where the loudpeaker i radiating into full pace (4π olid angle). Cabinet dimenion become comparable to wavelength at increaing frequency and the radiation of high-frequency wave become retricted to the half-pace (π olid angle) in front of the cabinet thu increaing the ound preure by 6 db. The increae in ound preure toward high frequencie i known a diffraction and will be introduced..4.. Circular baffle It i aumed that the direct ignal from the loudpeaker i radiated from a point ource into half pace in front of the loudpeaker cabinet (p, blue colour in Figure 5). At the edge of the cabinet i ome part of the wave propagated along the ide of the cabinet in order to fill the area behind the cabinet (p B, green). The dicontinuity at the edge give rie to a reflection of oppoite polarity (p, red). It will, for the cae of implicity, be aumed that the ound preure i halved at the dicontinuity o the backward radiation and the forward reflection are of equal amplitude but oppoite polarity. The effect of cabinet thickne will not be conidered. Baffle oudpeaker within a cloed cabinet Model uing the radial ymmetry oudpeaker eflection repreented a a ingle point ource r pb p/ irect ignal eflected ignal p p -p/ Obervation point Figure 5 A implified model auming the loudpeaker located at a circular baffle with negligible thickne (cloed cabinet). The reflection are aumed aembled into one point ource located at the edge of the urface. The ound preure at obervation ditance from the loudpeaker i repreented by a point ource with volume velocity Q S U, which i radiating into the half pace at the front of the loudpeaker: p iωq π ( ) exp( ik) ere i k ω/c πf/c the angular wave number. eflection from the cabinet boundary i delayed by ditance r, the radiu of the baffle, the ound preure i one-half that of the loudpeaker ound preure and the polarity i the oppoite in order to partially cancel the radiation from the loudpeaker outide the baffle. The attenuation due to the increaed path length will be ignored (ditance r i aumed mall compared to the obervation ditance ), and the reflected ound preure at the obervation point become: iωq p 4π ( ) exp( ik( r) ) Ørted TU Acoutical Technology 6

64 oudpeaker croover network The reulting far field ound preure on the axi of ymmetry i the um of the two expreion: Thi can be reduced to: iωq iωq p F π 4π ( ) exp( ik) exp( ik( r) ) p F iωq π ( ) exp( ik) exp( ikr) The firt two term are identified a p, the direct ound from the loudpeaker, o the far field ound preure with a circular baffle can be decribed a: p F ( ) exp( ikr) p ( ) At low frequencie, where the exponential approache unity, i the level one-half that of a loudpeaker within an infinite baffle, i.e. the level i 6 db down compared to the pecification in the data heet (meaured at π). At higher frequencie, where the exponential rotate within the unit circle, i the amplitude ocillating from.5 to.5 time the level with an infinite baffle (from 6 db to.5 db). Nominal level (. db) exp( ) Firt peak (.5 db) exp ( ) Firt dip (-6. db) exp( ) ikr kr arcco. 4 ikr kr π. 4 ikr kr π 6. 8 c f NOM. r c f PAK. 5 c f IP r r For a circular baffle with radiu r. m, the nominal level i croed firt time at 7 z, the firt peak i at.7 kz and the firt dip i at.4 kz. The behaviour of the equation within the quare bracket i hown in Figure 54. The ripple at higher frequencie will be le pronounced in real life ince the loudpeaker become directive thu reducing the effect of the reflection. Figure 54 iffraction for a loudpeaker located at the centre of a circular tube of radiu r. The model doe not account for loudpeaker directivity or baffle thickne. Ørted TU Acoutical Technology 64

65 oudpeaker croover network.4.. Sectional baffle Aume a loudpeaker baffle deigned from ection of circle, uch a the one in Figure 55, where four ection of 9 each are hown. The number of ection N i arbitrary, a are the radii r k of the ection, but the analyi aume that the ection are all covering the ame angle (ϕ k 6 /N). oudpeaker r ϕ 9 r ϕ 8 Baffle ϕ r ϕ 7 r4 Figure 55 oudpeaker baffle contructed of four circular ection in order to mear the amplitude ripple. The ound preure of the direct ignal from the loudpeaker i a before modelled by a point ource radiating into half pace: p iωq π ( ) exp( ik) The reflection are modelled by point ource, one for each ection, each with a ound preure of /N of the ound preure of the direct ignal and of oppoite polarity. iωq pn n,..., 4Nπ ( ) exp( ik( r )) n, N The reulting far field ound preure become: p F ( ) p ( ) p ( ) Inerting the expreion and collecting term: Thi i equivalent to: N n ω pf ikr n π N n N i Q ( ) exp( ik) exp( ) p F N N n ( ) exp( ikr ) p ( ) At low frequencie are all exponential approximating unity o the um equal N and the expreion within the quare bracket become one-half, o the initial level i 6 db compared to the infinite baffle. An example i hown in Figure 56 with four different value of the radii. The peak-peak ripple i reduced from 9 db for the circular baffle to db for kr < 5 by thi et-up. n n Ørted TU Acoutical Technology 65

66 oudpeaker croover network Figure 56 iffraction for a baffle coniting of four circular ection, each 9 wide and with radii r :r :r :r 4.4:.8:.:.6, which cale the reult o it i comparable to the previou plot..4.. Square baffle A quare baffle i hown in Figure 57, and differ from the previou analyi by the ditance from loudpeaker to edge being a function of angle ϕ. The ymmetry of the ytem can implify the analyi ince there are eight identical ection, which can be repreented by the ection from to π/4 and the reult can be repeated eight time. ϕ 45 Baffle oudpeaker ϕ 45 r Baffle r(ϕ) ϕ r Baffle d ϕ r tan(ϕ) ϕ -45 Figure 57 A quare baffle with ide length r being defined by polar coordinate. Symmetry divide the baffle into eight identical ection of 45 each. The ditance from the centre of the loudpeaker to the edge i given by Pythagora a: d r ( r ( )) co ϕ Uing in ϕ co ϕ, thi can be reduced to: r d ( ϕ) r ec co ( ϕ) ( ϕ) The free-field radiation from each of the eight ection at ditance become: p n iωq 8 4π π / 4 ( ) exp[ ikr ec( ϕ) ] dϕ, n,, K, 8 Factor one-eight i due to ymmetry and the reduced range of the angle. The ecant can be approximated by the Taylor erie expanion [Schaum.5]: 4 6 ϕ 5ϕ 6ϕ π ec( ϕ )..., ϕ <! 4 7 Ørted TU Acoutical Technology 66

67 oudpeaker croover network It i poible to terminate the erie after the econd term ince the third term i.8 for ϕ π/4 and the higher-order term are even maller. The ound preure become: p n ( ) iωq π π / 4 iωq exp π ϕ exp ikr dϕ π / 4 ( ikr) ϕ exp ikr dϕ The Taylor erie expanion of the exponential i [Schaum.5]: x x exp( x ) x..., x <!! The erie expanion converge for any x if the number of term i ufficiently large o the expanion i not an approximation but the erie will be terminated after integration ha been carried out. The integral become: π / 4 ikrϕ exp( ) dϕ π / 4 π / 4 ikrϕ kr i ϕ kr ϕ i ϕ 6 π kr π i i.5kr! ikrϕ ( kr) ( kr) 4 ϕ i ( kr) ( kr) 5 ( kr) π ( kr) 4 i.7854 i.87kr.75 8 ikrϕ! 6 ϕ K dϕ ϕ i ϕ K π / 4 π K 6 4 ( kr) i.5( kr) 7 K dϕ K The approximation i valid for vanihing higher-order term. Thi require that: ( kr) < kr < c f < πr For r.5 m (a baffle width of. m) the limiting frequency become f <.6 kz. The approximation i a phae vector with an amplitude of.79 and a phae of at low frequencie and an amplitude of.8 and a phae of at k. The change in amplitude can be neglected and the phae become:.5kr arctan arctan.79 (.9kr) Ørted TU Acoutical Technology 67

68 oudpeaker croover network ence, the integral can be repreented by the approximation: π / 4 ikrϕ exp( ) dϕ.79exp The ound preure from one of the edge become: p k iωq π iωq.494 exp π (.9ikr) ( ) exp( ikr).79 exp(.9ikr) The reulting far field ound preure become: Inerting the expreion: p F (.9ikr) ( ) p ( ) p ( ) 8 n iωq iωq p F 9 π π ( ) exp( ik).95 exp(. ikr) ence, the far-field ound preure for kr < : p F N k ( ).95 exp(.9ikr) p ( ) Thi i almot the ame equation a for the circular baffle; the amplitude of the reflection i reduced from.5 to.95 and the phae due to the delay i 9 % fater. Thi could indicate that the model i over-implified. n Figure 58 iffraction from a quare baffle with ide length r. The model i valid for kr <. The peak-to-peak ripple i reduced from 9.5 db (.5 db to 6. db) for the circular baffle to 7. db (.9 db to 4.4 db) with the quare baffle. Ørted TU Acoutical Technology 68

69 oudpeaker croover network.5. itening angle A loudpeaker ytem with a croover network output ignal from two loudpeaker, which are diplaced vertically or horizontally. itening at angle different from introduce a time delay between the loudpeaker cauing interference between the loudpeaker. The problem can be olved by arranging the loudpeaker in-line on the front plane but the problem remain for the other axi and will be analyed below..5.. Two loudpeaker Two loudpeaker are placed at the ame baffle with the ditance between centre. They are aumed ideal tranducer in the following to implify the analyi. ϕ > ϕ On axi ϕ ϕ < Figure 59 oudpeaker are located above each other to avoid horizontal time delay. Vertical offet of the litening angle introduce a time delay thu changing the phae difference between the loudpeaker at the umming poition. Offet angle ϕ introduce a ditance, which increae the litening ditance for one of the loudpeaker and reduce the ditance for the other. in For.5 m between the loudpeaker and ϕ 5 offet the ditance i mm. The ditance give rie to a time hift, which i poitive for the channel where the ditance i increaed and negative for the other. τ ϕ c c ( ϕ) in For.5 m and ϕ 5 the time delay i τ ϕ 95 µ. Time hifting i equivalent to phae. ( ϕ) ω k θ ωτ ϕ in in c ( ϕ ) ( ϕ) With angle ϕ poitive upward, and with top poition T and bottom poition B, the phae relation for the two channel become: T B exp exp ( iθ ) exp( ik in( ϕ) ) ( iθ ) exp( ik in( ϕ) ) Thi i ignificant for mot of the frequency range ince k <. i only atified for frequencie below z (.5 m) and litening angle above ±5 mut be expected. Ørted TU Acoutical Technology 69

70 oudpeaker croover network The reult i hown in Figure 6 for a firt-order croover with ±5 litening angle correponding to ±95 µ of time delay (.5 m loudpeaker ditance). In Figure 6 and Figure 6 are croover of econd and third order with ±5 and ± litening angle. The figure apply to poitive and negative angle a well. Figure 6 Amplitude repone with angle 5 (left) and 5 (right) for firt-order croover. Figure 6 Amplitude repone with angle ±5 (left) and ± (right) for econdorder croover with inverted treble. Figure 6 - Amplitude repone with angle ±5 (left) and ± (right) for fourthorder croover. Ørted TU Acoutical Technology 7

71 oudpeaker croover network The impact upon amplitude repone i ignificant o the loudpeaker hould be tilted to point toward the litener. Thi can be problematic for public-addre application, where the loudpeaker mut cover a large area. The calculation are equally valid for loudpeaker diplacement offet, where the loudpeaker are axially diplaced. Thi i hown to the right of Figure Three loudpeaker The phae difference between the ba and treble loudpeaker can be cancelled by uing two ba loudpeaker centred around the treble loudpeaker. - ϕ > ϕ On axi ϕ ϕ < Figure 6 Two ba loudpeaker balance the time delay to zero at low frequencie. Offet angle ϕ introduce a ditance, which increae the litening ditance for one of the ba loudpeaker and reduce the ditance for the other. The ditance between the ba loudpeaker i o the equation for the ditance become. ence the phae: ω θ in c in( ϕ) ( ϕ) k in( ϕ) The tranfer function for the output from the combined ba loudpeaker i: B exp ( iθ ) exp( iθ ) co( kin( ϕ) ) So, the phae angle i tranformed into an amplitude error, which i zero for on-axi litening and reduce the ba loudpeaker amplitude around croover. The amplitude error i a function of frequency, and ocillate for high frequencie. The firt null occur at the frequency where kin(ϕ) become π/, which i: c f N 4 in( ϕ) For.5 m i the firt null at f N. z at ϕ.5 and f N 69 z at ϕ.. The reult i hown in Figure 64 for the firt-order network, which now accept much larger litening angle. The reduction in ba loudpeaker amplitude i clearly een and the ripple in the high-frequency range i due to interference between the ba loudpeaker. Ørted TU Acoutical Technology 7

72 oudpeaker croover network Figure 64 Amplitude repone with angle ±5 (left) and ± (right) for firt-order croover. Figure 65 Amplitude repone with angle ±5 (left) and ± (right) for econdorder croover with inverted treble. Figure 66 - Amplitude repone with angle ±5 (left) and ± (right) for fourthorder croover. Ørted TU Acoutical Technology 7

73 oudpeaker croover network.6. Boundary reflection oudpeaker ued within room are affected by reflection from the urface, which interfere with the direct ignal cauing peak and dip in the amplitude repone. It i common to ue the ray method for the analyi of thi problem, ee Figure 67. The ound ray from the loudpeaker i reflected by the urface and echoed back into the room a if the urface were a mirror. The reultant ound preure i calculated by addition of the individual ound ray from the loudpeaker and the virtual ource. The ray method aume that the location of ource and obervation point are in the free field, thu ignoring near-field effect uch a the increae in radiation impedance of two coherent ource located cloe to each other. Thi limit the validity of the ray method to frequencie where the ditance between ource, urface and obervation point mut all be large compared to wavelength..6.. One reflecting urface A imple model will introduce the ray method before a more complete model i developed. Only one boundary i preent, typically the floor or one of the wall. The loudpeaker i aumed pointing directly at the litener at horizontal ditance from the loudpeaker. The litener i ditance from the boundary and the loudpeaker i ditance from the boundary. The boundary i aumed rigid o that all ignal i reflected and the loudpeaker directivity i decribed a defined previouly by angle θ. oudpeaker θ Obervation point θ Surface Mirror θ θ Figure 67 A implified model with one reflecting boundary. irect ignal path length i and reflected ignal path length i. The loudpeaker i aumed pointing to the litener. The direct ound at the obervation point, located at ditance from a monopole ource with volume velocity Q, i: p iωρq π ( ik ), where ( ) exp 4 The reflection i delayed due to the increaed path length and become: p iωρq ( ik ) ( ), where ( ) exp θ 4π The loudpeaker radiation toward the boundary i dependent upon the angle θ, which can be decribed a the um of two angle according to the drawing at the right ide of Figure 67: θ θ θ arctan arctan Ørted TU Acoutical Technology 7

74 oudpeaker croover network The reultant ound preure become: Thi can be written: p iωρq iωρq exp exp 4π 4π p ( ik ) ( ik ) ( θ ) ( θ ) exp ( ik( )) p The effect of boundary reflection i to multiply the direct ound preure by a complex phaor with an amplitude between and and a phae, which i a function of frequency and ditance. At low frequencie i the exponential function approaching unity, o the amplitude i increaed, with a maximum of time (6 db) the direct ignal. At higher frequencie will the amplitude ripple between a low value where the exponential i and a high value where the exponential i. The firt null occur at: k ( ) ω ( ) c π f NU ( ) For m and.8 m the firt null i at f NU z. The firt peak occur at two time f NU, which i at 4 z. The reult i hown in Figure 68 for two different et-up. The left hand figure i with both loudpeaker and litener at m height and m between loudpeaker and litener. The reflected ignal i delayed.8 m and the firt null occur at z. The ripple amplitude decay at high frequencie ince the loudpeaker become directive. The right hand figure i with the loudpeaker at. m ditance from the floor. The reflected ignal i now only delayed.7 m o the firt null occur at 96 z and loudpeaker directivity remove mot of the ripple. c Figure 68 Amplitude repone with loudpeaker at m height (left) and. m (right), with m litening ditance and m litener height. oudpeaker radiu wa. m. Ørted TU Acoutical Technology 74

75 oudpeaker croover network.6.. ectangular room A more involved model will be developed, which included up to ix reflecting boundarie decribing a conventional rectangular room. z Mirror (-x,y,z) oudpeaker (x,y,z) oudpeaker (x,y,z) Mirror (x,-y,z) (,,) θ x y x Mirror (x,y,-z) Obervation (x,y,z) Figure 69 oudpeaker i at location (x,y,z ), direction i along the x-axi and the obervation point i at location (x,y,z ). Mirror ource i located behind the loudpeaker at (-x,y,z ) ince the x-axi i normal to the yz-urface, and o forth for the other reflection and. eflection boundarie at (x,y,z ) introduce a mirror ource at 4 (x x,y,z ) and o forth for reflection 5 and 6. eflecting boundarie are aumed located a the ide of a rectangular box with one corner at (,, ) and another at (x, y, z ), thu defining the ize of the room. The ix urface will be referenced through a boundary number b to 6, a lited below. Boundary, b Surface Comment Y-plane with x Wall behind loudpeaker X-plane with y eft wall XY-plane with z Floor 4 Y-plane with x x Wall behind litener 5 X-plane with y y ight wall 6 XY-plane with z z oof With the loudpeaker and obervation point located a hown and auming that the coordinate (x n, y n, z n ) are all within the room ( < x n < x, and o forth), the path P n from loudpeaker n to the litener i defined a the ditance from the ource location vector n to the obervation point location vector : Index n i the loudpeaker number. P ( x x y y z z ) n n n n eflected ignal path are repreented a P nb for a path from loudpeaker n reflected through boundary b to the litener. The reflection path vector for reflection in urface with one corner at (,, ) are: P P P n n n ( x xn y yn z zn ) ( x xn y yn z zn ) ( x x y y z z ) n n n n Ørted TU Acoutical Technology 75

76 oudpeaker croover network And for reflection in urface with one corner at (x, y, z ): P n4 P n5 P n6 ( x ( x xn ) y yn z zn ) ( x xn y ( y yn ) z zn ) ( x x y y z ( z z )) n oudpeaker radiation i dependent upon the angle from on-axi o the direction of the loudpeaker mut be pecified. oudpeaker n i orientated with the main direction (loudpeaker front ide) pointing along the direction vector n : n n ( x y z ) n The coordinate can be determined from the horizontal angle θ, which i along the x- axi and the vertical angle φ, which i at the xy-plane (z ), a: x y z n n n co in in n n ( θ n ) co( φn ) ( θ n ) co( φn ) ( φ ) For loudpeaker pointing along the x-axi the vector become (,, ). The obervation angle i different for the reflection; one coordinate change ign due to the reflection. n n n n4 n5 n6 n ( xn yn zn ) ( xn yn zn ) ( x y z ) n The obervation angle for the direct ound from the loudpeaker θ n i determined a the angle between the loudpeaker direction vector n and the vector pointing from the loudpeaker to the obervation point P n. The angle i for the loudpeaker pointing directly at the obervation point. The obervation angle i extracted from the definition of the inner product (dot product) between n and P n [5]: n P n n P n n co n ( θ ) The obervation angle for the direct ignal from loudpeaker n become: θ n n P arcco n P The inner product i defined in MATAB with A B expreed a A * B, where A mean the tranpoed of column vector A. The obervation angle for the reflection are: θ θ n n4 arcco n arcco n n4 4n P n P n Pn 4 P n4 θ n θ n5 arcco n arcco n n5 n5 nn n Pn P n n Pn 5 P n5 θ n n θ n6 arcco n arcco n n6 n6 P n P n Pn 6 P n6 Ørted TU Acoutical Technology 76

77 oudpeaker croover network The ound preure at the obervation point from loudpeaker n and ix reflection, auming equal volume velocity Q for all loudpeaker and mirror ource, i: p n iωρq exp ( ikpn ) ( θ n ) exp( ikpnb ) Cb( θnb ) 4πPn b 6 iωρq 4πP The reflection coefficient C wa included for two reaon; it i an eay way to elective enable and diable reflection and it allow ue of partially reflective urface. Set all coefficient to zero to completely cancel reflection or et one or more to unity to electively enable reflection. Value between zero and unity imulate panel aborber, thick curtain or wall with large opening and the contant may be complex, if required. (θ) repreent the loudpeaker directivity, which i defined in.. The reultant ound preure from N loudpeaker with ix reflecting urface: iωρq p 4π ( θ ) N 6 n exp( ikpn ) n Pn b nb Cb P ( θ ) nb nb exp ( ikp ).6.. ome entertainment A typical et-up for home entertainment could be with the loudpeaker m above ground and the litener at m ditance and with the head at the ame height a the loudpeaker. nb m m Figure 7 Set-up for home entertainment with a loudpeaker in a living room, which wa 5 m m and m high in the analyi. The amplitude repone i hown in Figure 7, left. The reflection i delayed.8 m correponding to a half wave length at z where detructive interference occur. ower frequencie are increaed due to the reflection being in-phae and higher frequencie ocillate. An improvement for the low-frequency operation can be obtained by reducing the loudpeaker elevation a hown to the right. The curve are imilar to Figure 68, which wa calculated uing the implified model with one reflecting boundary. The difference in Figure 7, right, i due to the loudpeaker pointing along the x-axi and not directly toward the litener. Ørted TU Acoutical Technology 77

78 oudpeaker croover network Figure 7 Amplitude repone for loudpeaker at m height and m ditance (left) and height reduced to. m (right). itening height wa m. iaphragm radiu wa a. m and the loudpeaker point along the x-axi. The ocillation ceae at high frequencie due to the frequency-dependent loudpeaker directivity and the mall burt of ocillation at high frequency i caued by ide lobe. Anyway, the model i not valid for high frequencie where the diaphragm i not vibrating a a rigid piton, the limit i around ka <, which i.6 kz for a diaphragm diameter of a. m. Activating reflection from the ide wall with m ditance from the loudpeaker and litener reintroduce the reflection with the dip at z but the dip i le pronounced now ince the overall level i increaed. Moving the loudpeaker cloer to the wall increae the frequency of the dip. Figure 7 Amplitude repone for loudpeaker at. m height, m from a wall and at m litener (left). itance to wall reduced to.5m (right). The high-frequency repone i more ragged than before ince three ignal path are combined (direct ound and two reflection). Activating the reflection from the wall behind the loudpeaker increae the low-frequency level. Ørted TU Acoutical Technology 78

79 oudpeaker croover network Figure 7 Amplitude repone for loudpeaker at. m height, m from both wall and at m to litener (left). itance to all three urface reduced to. m (right). A decent amplitude repone i obtained with the loudpeaker located cloe to both the corner between floor and wall a hown in Figure 7, right. The amplitude repone indicate the need of croover to the midrange loudpeaker below z o the corner i optimal for a ub woofer (ignoring the tanding wave within a real room). The midrange and treble loudpeaker hould be diplaced from the ub woofer in order to reduce the effect of room reflection. All urface are activated in Figure 74 where the loudpeaker i located at ome ditance from the corner at the left drawing and i moved cloer to the corner in the econd drawing. The mot flat amplitude repone i obtained cloe to the corner, but uing the corner poition may caue excitation of room reonance, not modelled here, and thi may lead to a booming reproduction of low frequencie. Figure 74 Amplitude repone for a room of 5 m by m with m height. oudpeaker at m ditance from corner (left) or. m from corner (right). It i intereting to note, that the level at low frequencie i not increaed db for each urface, a i ometime tated in the popular literature; the figure are rather 6 db from the firt urface, db from the two next, and an inignificant amount from additional urface. There i no imple explanation to thi obervation it could be expected that the direct and reflected ignal would add almot lole at low frequencie, but it appear not to be o. Ørted TU Acoutical Technology 79

80 oudpeaker croover network.6.4. Public addre Another application i a large room for theatre or concert, which i fitted with a loudpeaker ytem for public addre. m m Figure 75 arge room for concert, with dimenion 5 m m and 5 m high. The loudpeaker i located 5 m from each wall and m above the floor and the litening poition wa 5 m from the loudpeaker, / from the wall and m above the floor. The urface were aborbing. The reulting amplitude repone for a room with reflecting urface i hown in Figure 76 for a room without damping material. Only the direct ignal and ix reflection are included but the amplitude repone i very ragged with ± db variation within the main frequency range. The effect of adding damping to the urface i hown to the right, where the amplitude repone i within ±5 db for mot of the audible range but the overall level i reduced ome db ince the reflection are reduced in level. Figure 76 Amplitude repone for a room without damping (left) and the reflection coefficient adjuted to. for the wall behind loudpeaker and the left and right wall and. for roof, floor and the wall behind the litener (right). The decaying high-frequency amplitude i caued by the directivity of the loudpeaker; the litener i off-axi. Ørted TU Acoutical Technology 8

81 oudpeaker croover network.7. oudpeaker characteritic Four loudpeaker have been elected a being repreentative for the loudpeaker ize found in two-way and three-way ytem and are lited in Table together with the publihed parameter of interet for thi tudy. The value are not directly ued in thi document but have formed the bai for election of parameter. Table Typical loudpeaker characteritic for two-way and three-way ytem. oudpeaker model Symbol Woofer Ba Midrange Treble Unit P:Peerle, S:ScanSpeak S:6W8667 P:5W S:M866 S:T95 iaphragm diameter 68 mm ffective diaphragm area S m Voice coil reitance Ω Voice coil inductance m Force factor B N/A iaphragm & voice coil ma M M g Supenion compliance C MS mm/n Supenion friction lo MS kg/ Senitivity (.8 V) ref db inear excurion x MAX ±9. ±5.5 ±.5 ±.4 mm lectrical quality factor Q MS Mechanical quality factor Q S Total quality factor Q TS quivalent volume V AS m eonance frequency f S z Frequency for ka f A kz Frequency due to voice coil f C kz The top value refer to the publihed figure for the tated loudpeaker. The bottom two value are calculated uing the equation from thi ection. Ørted TU Acoutical Technology 8

82 oudpeaker croover network.8. Group delay Group delay pecify the delay experienced by a group of inuoidal component, which have frequencie within a narrow frequency interval about f ω/π. The bandwidth in thi interpretation mut be retricted to a frequency interval over which the phae repone i approximately linear..8.. Calculation method The group delay i defined a the rate of change of phae with repect to frequency [7]: τ G dθ dω The phae i a property of the tranfer function (ω), which can be written in polar notation with G(ω) a the amplitude repone and θ(ω) a the phae repone where both are real valued function: ( ω) G( ω) exp( iθ ( ω) ) Thi can be eparated into amplitude and phae term uing the logarithmic function. ln ( ( ω) ) ln( G( ω) ) iθ ( ω) Thi i differentiated with repect to angular frequency: d ln dω d dω d dω ( ( ω) ) ln( G( ω) ) i θ ( ω) Uing the differentiation rule for the logarithm (Schaum.7): '( ω) ( ω) G G '( ω) ( ω) iθ ' G (ω) and θ (ω) denote the derivative of G(ω) and θ(ω) repectively. The group delay i repreented by the imaginary part, and ince G (ω)/g(ω) i real thi become. τ G θ ' ( ω).8.. Implementation in MATAB Im ( ω) '( ω) ( ω) The derivative of (ω) can be expreed a the lope of (ω) within a narrow frequency range from ω to ω ω by ue of the definition of the derivative (Schaum.): ' ( ω) lim ω ( ω ω) ( ω) ω The difference ω cannot reach zero o a finite value of. will be ued. The frequency variable i iω/ω if/f o the increment become ω.ω or alternatively f.f. The incremental frequency i thu z for a normaliation frequency (croover frequency) of f z. Ørted TU Acoutical Technology 8

83 oudpeaker croover network ence, calculation of (ω) uing the following expreion: τ G θ ' ( ω) Im ( ω ω) ( ω) ( ω) ω The frequency i defined a the vector f *[.:.:], uing MATAB notation, o the incrementally larger frequency ue another vector g defined from f to output the tep following immediately after, i.e. defined a g f.* f..8.. Verification Two filter will be analyed for teting the MATAB implementation of group delay. The firt tet object i a firt-order low-pa filter, which i brought onto a form uitable for analytic extraction of the phae. The phae i: iω ω i ω ω iω iω iω ω ω ω ω ω Im θ arctan e And the group delay become (Schaum.): τ GP { } i { } ω ω d ω i dω ω ω ω ω The reult i hown in Figure 77 and indicate good agreement. Figure 77 Group delay uing the analytic expreion (left) and the calculation from the tranfer function of the low-pa filter (right). Ørted TU Acoutical Technology 8

84 oudpeaker croover network Ørted TU Acoutical Technology 84 The econd tet object a econd-order all-pa filter with contant amplitude for a. a The derivative of the tet function can be calculated from the definition of the derivative of a quotient of two function (Schaum.9): ( ) ( ) ( ) ( ) ( ) ' a a d d d d a d d ω ω ω The derivative of the frequency variable: ( ) ( ) i i i i d d d d i i d d d d ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ence, the derivative of the tet function: ( ) ( ) ( ) ' a i ia i a ω ω ω Thi wa implemented in MATAB for comparion and the reult of the analytical expreion i hown in Figure 78 with the calculation method baed on tranfer function output hown in Figure 79. Figure 78 Group delay uing a tet function with ymbolic differentiation carried out before calculation. Plot for a (left) and a (right).

85 oudpeaker croover network Figure 79 Group delay uing the tranfer function output at two frequencie with incremental ditance (i.e. f and f f). Plot for a (left) and a (right). There are no viual difference between the figure, o the agreement i good. But a couple of figure uing ome few tranfer function cannot proof the validity of the calculation but it indicate that the method come cloe on a couple of tet. Ørted TU Acoutical Technology 85

86 oudpeaker croover network 4. Aembling the model It i time to combine the model into a ytem deign trial. The target i a low-budget two-way ytem, o the deign mut be kept imple and uncomplicated to aemble during production, i.e. uing a imple paive croover network. 4.. oudpeaker model The loudpeaker ytem i a cloed cabinet with baffle meaure hown in Figure 88. The loudpeaker unit are: A ba loudpeaker, 8 inch with 7 z reonance frequency and a total quality factor of.7 within the cabinet and it i aumed to reproduce moothly to kz at db with a cut-off lope of 6 db/octave at higher frequencie. A treble loudpeaker, inch dome with kz reonance frequency, a total quality factor of and high-frequency roll-off above kz at db. The amplitude and phae repone of the ytem are hown in Figure 8 where the total curve repreent a ytem where the output are added without a croover filter. Figure 8 oudpeaker model for the two-way ytem. The ba loudpeaker i defined by the reonance frequency f S 7 z, total quality factor Q TC.7 and highfrequency roll-off at f C kz. For the treble loudpeaker i f S kz, Q TC. and f C kz. 4.. Croover network A croover network mut protect the treble loudpeaker, and thi i in thi report aumed fulfilled for at leat db of attenuation at the reonance frequency of kz. Uing a econd-order croover network offer 4 db of attenuation with a croover frequency of 4 kz, o thi will be ued for a tart. A notch at the croover frequency i to be expected uing a econd-order croover network due to 8 of phae difference between the ba and treble loudpeaker o the treble loudpeaker i inverted. Auming ideal addition of the output require a croover network with 6 db attenuation at the croover frequency o the filter mut ue a quality factor of.5 ince the attenuation i log (Q) db. The reulting amplitude and phae repone are hown in Figure 8. A dip of approximately db i een at the croover frequency. Ørted TU Acoutical Technology 86

87 oudpeaker croover network Figure 8 eultant amplitude and phae repone for the ytem with a econdorder croover network at 4 kz and inverted treble. The phae difference between ba and treble i around 5, o the loudpeaker are not in-phae due to the phae repone of the loudpeaker unit. The difference can be reduced by removing one of the pole from the low-pa filter thu implifying the croover network by reducing the low-pa filter to firt order. The reult i hown in Figure 8, where the phae difference i cloe to at 4 kz and the amplitude repone i improved. Figure 8 oudpeaker repone for a ytem with a firt-order low-pa filter for the ba loudpeaker and econd-order filter for the treble loudpeaker. A croover network with an inductor in erie with the loudpeaker i not attractive, firt of all due to the loudpeaker impedance, which i increaing at high frequencie and thu oppoing the intended low-pa filtering, and econd, becaue the inductor i a relatively cotly component. CT Ba B T Treble T Figure 8 Paive croover network for the two-way ytem. The ba loudpeaker high-frequency roll-off i ued a the low-pa channel of the croover network. Ørted TU Acoutical Technology 87

88 oudpeaker croover network If the inductor i removed, the reulting croover network conit of a econd-order high-pa filter for the treble loudpeaker and no filter for the ba loudpeaker. Thi i very attractive from a production point of view, but the reulting repone in Figure 84 (left picture) how repone peaking ome db around the croover frequency. Thi peak will not be removed by the inductor, unle an impedance compenation i included for the ba loudpeaker, and thi i not the idea behind the ytem. Figure 84 oudpeaker repone with the low-pa filter removed (left) and with the treble loudpeaker attenuated 6 db (right). Neither deign are acceptable. The peak can be removed by attenuation of the treble loudpeaker, but the reultant amplitude repone, hown in Figure 84 (right picture), how that the reult i an early roll-off in the treble, which cannot be accepted. The phae repone of the ba channel hould include two pole, one from the loudpeaker and another from the croover network, but the current deign only include the pole from the loudpeaker. The phae i not ufficiently negative, o it could be poible to add the miing phae by time-hifting of the ignal from the treble loudpeaker by moving it axially. A time hift introduce a rotating phae where the angular peed i defined by frequency f and the ditance the treble loudpeaker i moved, according to: TIM exp ( iτ f ), τ c A delay of µ correpond to a ditance of 4 mm and the reult of thi movement of the treble loudpeaker i hown to the left in Figure 85. The reult i not the intended removal of the peak, rather i the amplitude repone completely ruined by the movement. oing the oppoite, advancing time by moving the treble loudpeaker toward the litener, i hown to the right i Figure 85. There i a little change but the peak i not at all removed. So, it eem that we have to live with the db boot around 4 kz. Ørted TU Acoutical Technology 88

89 oudpeaker croover network Figure 85 oudpeaker repone with loudpeaker delayed µ by moving i 4 mm into the cabinet, or the ba loudpeaker 4 mm out from the cabinet (left) and advancing the time µm by moving it the other way (right). 4.. Angular repone The loudpeaker proved rather enitive to modet delay, thu indicating that it would be a good idea to tudy the behaviour to off-axi litening. Two angle are relevant for thi analyi, the horizontal angle repreenting a loudpeaker pointing toward one of the ide of the head of the litener, and the vertical angle repreenting a loudpeaker pointing above or below the head. The effect of horizontal angle i hown in Figure 86 for two angle at 5 and and how that the amplitude repone i omewhat enitive to change in the horizontal angle. The ba loudpeaker i the problem ince the diaphragm diameter i comparable to wavelength around the croover frequency. Figure 86 oudpeaker repone for a horizontal angle of 5 (left) and (right). The loudpeaker radiu wa mm for the ba loudpeaker and mm for the treble loudpeaker. The ba loudpeaker become directive above 55 z (ka ). The effect of vertical angle i hown in Figure 87 for two angle at 5 and and how that the amplitude repone i quite enitive to change in the horizontal angle. The loudpeaker i not ymmetrical on the vertical axi, and the behaviour for negative angle i not hown here, but the curve indicate that the loudpeaker mut be tilted o that it point to the litener. Ørted TU Acoutical Technology 89

90 oudpeaker croover network Figure 87 oudpeaker repone for a vertical angle of 5 (left) and (right). A concluion o far i that the peak at the croover frequency hould be kept a it i although the on-axi repone could be improved. The conequence of modifying the loudpeaker to a flatter amplitude would mot probably be degradation of the off-axi repone. The direct ound will be lightly improved around the croover frequency but the off-axi repone how a decreae in the ame range and thi affect the ound preure level within the reverberant field eflection eflection from the edge of the loudpeaker interfere with the loudpeaker and wa modelled in ection.4 where the model with circular ection were mot ucceful and will be ued for the analyi below. The loudpeaker cabinet i rectangular and will be modelled by four circular ection with radii equal to the mean ditance from the centre of the ba loudpeaker to the edge. Thi i a coare model, but it give an idea of the effect of the front baffle. 5 5 mm 4 9 mm 5 mm mm 5 mm 4 mm 5 mm mm W 5 mm Figure 88 oudpeaker front baffle meaure for calculation of diffraction. The hortet mean ditance i the average between and ; hence.8 m. ue to ymmetry i the two next hortet ditance identical and the average between, and 4 ; hence. m. The larget mean ditance i the average between 4 and 5 ; hence.7 m. The reult i hown in Figure 89 and i een to overrule the effect of the peak around the croover frequency. Alo een i the reduced output at low frequencie where the baffle become mall compared to wavelength. The loudpeaker i at low frequencie radiating into 4π olid angle, while the baffle limit the radiation angle to π at higher frequencie. Ørted TU Acoutical Technology 9

91 oudpeaker croover network Figure 89 oudpeaker repone with reflection from the baffle (left) and with reflection from one boundary included (right). The loudpeaker wa located m above ground at m ditance from the litener. eflection from a large urface (the floor) i hown to the right of Figure 89 with the loudpeaker m above ground at m ditance to the litener with hi or her ear m above ground. The low frequency repone i improved but a dip i een at z, which i due to the delayed ditance through the reflection path, which i.8 m longer than the direct path and caue detructive interference with the direct ignal. Interference become contructive at 4 z and o forth. Figure 9 oudpeaker at.5 m (left) and.5 m (right). The model include the croover network, the loudpeaker, diffraction and one reflection. An improvement i obtained by reducing the loudpeaker height to.5 m above the floor (Figure 9, left picture). The firt dip occur at 5 z and the firt peak i located at 7 z. The peak i of large amplitude but can be reduced by lowering the height to.5 m above the floor (Figure 9, right picture). The high-frequency ripple i due to the implified model and will be le pronounced in real life Concluion The model form an effective tool for initial loudpeaker deign although improvement are required epecially for the diffraction and boundary reflection model. Ørted TU Acoutical Technology 9

92 oudpeaker croover network 5. eference 5.. Book []: Brian C. J. Moore An introduction to the Pychology of earing, 5 th edition, 4, levier Academic Pre. []: W: Marhall each, Jr. Introduction to lectroacoutic & Audio Amplifier eign, rd edition,, Kendall/unt Publihing Company. []: eo. Beranek, Acoutic, Acoutical Society of America, 99 edition, 996. [4] Finn Jacoben, Peter Juhl adiation of Sound, not publihed, TU, 5. [5] an bert lektronik Ståbi, Teknik Forlag, 995. [6] Paper Paper: S. inkwitz, 'Active Croover Network for Non-Coincident river,' J. Audio ng. Soc., vol. 4, pp. -8 (Jan./Feb. 976) together with hi co-worker u iley 5.. ink - A very nice introduction to the deign of a croover network, including compenation of loudpeaker impedance, voice coil temperature and much more. Ørted TU Acoutical Technology 9

93 oudpeaker croover network 6. Appendix 6.. Plot tranfer function Amplitude and phae repone are plotted uing the following MATAB hell cript. epreentative function addreed within the cript are included in the following ection Main cript The file wa executed by entering plot_tranfer_function at the MATAB prompt. % Plot tranfer function. clear all % emove old tuff. % % Contant. % % Frequency. FBG ; % Start frequency (z). FN ; % Stop frequency (z). FSTP ; % Frequency increment (z). f [FBG:FSTP:FN]; % Frequency f. g f FSTP; % Frequency g (required for group delay). % Acoutical. F 4; % Croover frequency (z). c 4; % Speed of ound (m/)..5; % iplacement between loudpeaker (m). % Micellaneou. pi/8; 8/pi; TU logical(); FAS logical(); % Converion from degree to radian. % Converion from radian to degree. % Ued by the IF tatement. % % Croover network coefficient. % % Second order, N : A.; %. % Third order, N : % A.; % A.; % Fourth order, N 4: % A.8; %.,.6 % A 4.; %.,.4 % A.8; %.,.6 % Sixth order, N 6: % A.; % A.; % A 6.; % A4.; % A5.; % % Croover network at frequency f. % ba_f ; % treble_f ; Ørted TU Acoutical Technology 9

94 oudpeaker croover network % ba_f lowpa_(f,f); % treble_f highpa_(f,f); % ba_f lowpa_(f,a,f); % midrange_f bandpa_(f,a,f); treble_f highpa_(f,a,f); % ba_f lowpa_(f,a,a,f); % midrange_f bandpa_(f,a,a,f); % treble_f highpa_(f,a,a,f); % ba_f lowpa_4(f,a,a,a,f); % midrange_f bandpa_4(f,a,a,a,f); % treble_f highpa_4(f,a,a,a,f); % ba_f lowpa_6(f,a,a,a,a4,a5,f); % treble_f highpa_6(f,a,a,a,a4,a5,f); % % Croover network at frequency g. % ba_g ; % treble_g ; % ba_g lowpa_(f,g); % treble_g highpa_(f,g); % ba_g lowpa_(f,a,g); % midrange_g bandpa_(f,a,g); treble_g highpa_(f,a,g); % ba_g lowpa_(f,a,a,g); % midrange_g bandpa_(f,a,a,g); % treble_g highpa_(f,a,a,g); % ba_g lowpa_4(f,a,a,a,g); % midrange_g bandpa_4(f,a,a,a,g); % treble_g highpa_4(f,a,a,a,g); % ba_g lowpa_6(f,a,a,a,a4,a5,g); % treble_g highpa_6(f,a,a,a,a4,a5,g); % % Include loudpeaker model. % % Treble loudpeaker. FCT e; % Voice coil cutoff pole frequency (z). FT e6; % Voice coil cutoff null frequency (z). FST e; % Mechanical reonance frequency (z). QTCT ; % Total quality factor. treble_f loudpeaker(fct,ft,fst,qtct,treble_f,f); treble_g loudpeaker(fct,ft,fst,qtct,treble_g,g); % Midrange loudpeaker. FCM e; % Voice coil cutoff pole frequency (z). FM e6; % Voice coil cutoff null frequency (z). FSM ; % Mechanical reonance frequency (z). QTCM ; % Total quality factor. % midrange_f loudpeaker(fcm,fm,fsm,qtcm,midrange_f,f); % midrange_g loudpeaker(fcm,fm,fsm,qtcm,midrange_g,g); % Ba loudpeaker. FCB e; % Voice coil cutoff pole frequency (z). FB e6; % Voice coil cutoff null frequency (z). FSB 7; % Mechanical reonance frequency (z). QTCB.7; % Total quality factor. ba_f loudpeaker(fcb,fb,fsb,qtcb,ba_f,f); ba_g loudpeaker(fcb,fb,fsb,qtcb,ba_g,g); % % Introduce directivity. % TTA ; AT e-; AM 5e-; AB e-; % oriontal or vertical ongle (degree). % Treble diaphragm radiu (m). % Midrange diaphragm radiu (m). % Ba diaphragm radiu (m). Ørted TU Acoutical Technology 94

95 oudpeaker croover network % treble_f directivity(tta,at,f).*treble_f; % treble_g directivity(tta,at,g).*treble_g; % midrange_f directivity(tta,am,f).*midrange_f; % midrange_g directivity(tta,am,g).*midrange_g; % ba_f directivity(tta,ab,f).*ba_f; % ba_g directivity(tta,ab,g).*ba_g; % % Introduce phae due to vertical angle. % % Vertical offet angle. VA ; % Vertical offet angle (degree). % Two-loudpeaker arrangement (ba treble). % treble_f exp( i**pi*f**in(va*)/(*c)).*treble_f; % treble_g exp( i**pi*g**in(va*)/(*c)).*treble_g; % ba_f exp(-i**pi*f**in(va*)/(*c)).*ba_f; % ba_g exp(-i**pi*g**in(va*)/(*c)).*ba_g; % midrange_f midrange_f; % midrange_g midrange_g; % Three-loudpeaker arrangement (ba treble ba). % treble_f treble_f; % treble_g treble_g; % ba_f (exp( i**pi*f**in(va*)/c)... % exp(-i**pi*f**in(va*)/c)).*ba_f/; % ba_g (exp( i**pi*g**in(va*)/c)... % exp(-i**pi*g**in(va*)/c)).*ba_g/; % % eultant tranfer function. % ATT.; Y e-6; % Attenuator network. % elay (). % um_f ba_f treble_f; % Two-way. % um_g ba_g treble_g; um_f ba_f - ATT*treble_f; um_g ba_g - ATT*treble_g; % Two-way, inverted. % um_f ba_f - treble_f.*exp(-i*f*y); % Two-way, time hifted. % um_g ba_g - treble_g.*exp(-i*f*y); % um_f ba_f midrange_f treble_f; % Three-way. % um_g ba_g midrange_g treble_g; % um_f one(ize(f)); % ummy (unity um). % um_g one(ize(g)); % % Include diffraction model. % um_f diffraction_ectional(.8,.,.,.7, f).*um_f; um_g diffraction_ectional(.8,.,.,.7, f).*um_g; % % Include one-dimenional boundary reflection model. % ;.;.; % oudpeaker ditance above floor (m). % itener ditance above floor (m). % orizontal ditance to litener (m). qrt(( - )^ ^); qrt(( )^ ^); T *(atan(( - )/) atan(( )/)); um_f boundary_imple(,,t,ab,f).*um_f; % % Plot amplitude pectrum. Ørted TU Acoutical Technology 95

96 oudpeaker croover network % figure(); emilogx(f, *log(ab(um_f)),... f, *log(ab(treble_f)),... f, *log(ab(ba_f))); legend('total','treble','ba'); axi([fbg FN - 5]); ylabel('amplitude (db)'); xlabel('frequency (z)'); % % Plot phae pectrum. % figure(); emilogx(f, *angle(um_f),... f, *angle(treble_f),... f, *angle(ba_f)); legend('total','treble','ba'); axi([fbg FN - ]); ylabel('phae ( )'); xlabel('frequency (z)'); % % Plot group delay pectrum. % % Group delay -Im('/). % groupdelay -imag((um_g - um_f)./(um_f*fstp)); % figure(); % emilogx(f, groupdelay); % axi([fbg FN e-]); % ylabel('group delay ()'); % xlabel('frequency (z)'); 6... Filter function All filter, being low-pa, band-pa or high-pa, are written in the below tyle. Input f i the centre frequency of the filter and A and o forth are coefficient to the polynomial and are pecified from the main cript file. Input f i a frequency vector, either from. to in tep of. (normalied frequency) or from z to z in tep of z. ifferent filter flavour are pecified for everal filter but only one i enabled by un-commenting the relevant definition. % Second order low-pa filter. function out lowpa_(f,a, f); (i/f)*f; % out ( A*)./( A*.^); % out ( (A/)*)./( A*.^); out./( A*.^); Only the lat equation i enabled oudpeaker A loudpeaker i defined by a econd-order high-pa filter and a firt-order low-pa filter. Input coefficient are pecified from the main cript. Variable in i the input repone, which i returned with the loudpeaker tranfer function. Ørted TU Acoutical Technology 96

97 oudpeaker croover network % oudpeaker tranfer function. function out loudpeaker(fc, F, FS, QTC, in, f); % FC Voice coil pole frequency due to and (z). % F Voice coil null frequency due to eddy current (z). % FS Mechanical reonance frequency (z). % QTC Total quality factor. out ((i*f/f)./(i*f/fc)).*... ( ((i*f/fs).^)./(i*f/(fs*qtc)(i*f/fs).^) ).*in; irectivity irectivity i defined by the Beel-function beelj, which i of the firt kind. The iftatement avoid diviion by zero at low frequencie and zero angle. function out directivity(t, A, f) % T Angle (degree). % A oudpeaker radiu (m). % f Frequency (z). TT (pi/8)*t; ka *pi*f*a/4; out one(ize(f)); % Convert angle from degree to radian. % Angular wave number. if (ab(tt) > ep) out ab(*beelj(,ka*in(tt))./(ka*in(tt))); end iffraction A implified model i ued, which i baed upon the circular ection method, here limited to four ection (9 each). function out diffraction_ectional(,,,4, f) % adiu of ection (m). % adiu of ection (m). % adiu of ection (m). % 4 adiu of ection 4 (m). ik i**pi*f*/4; ik i**pi*f*/4; ik i**pi*f*/4; ik4 i**pi*f*4/4; out -( exp(-ik)exp(-ik)exp(-ik)exp(-ik4) )/8; Boundary reflection A implified model i ued for reflection from the boundary, which i only uing one reflecting urface. The model include loudpeaker directivity and aume that the loudpeaker i pointing directly toward the litener. function out boundary_imple(,,t,a,f); % P irect path length (m). Ørted TU Acoutical Technology 97

98 oudpeaker croover network % P eflected path length (m). ik i**pi*f/4; out (/)*directivity(t,a,f).*exp(-ik*(-)); 6.. Plot boundary reflection Thi i the full cript for calculation of the effect of boundary reflection. The model aume a rectangular room with ix reflecting urface. The initial part of the cript check for negative or too large coordinate and zero diaphragm diameter. % Compute the reultant ound preure for a loudpeaker within a room. clear all; % % Input parameter. % % oom dimenion (m): X 5; Y ; 5; % Obervation point (m): X 4; Y ; ; %oudpeaker (m): X 5; Y 5; ; A ; % oudpeaker horiontal angle (degree). AV ; % oudpeaker vertical angle (degree). C.; % eflection coefficient for wall behind loudpeaker (m). C.; % eflection coefficient for left wall (m). C.; % eflection coefficient for floor (m). C4.; % eflection coefficient for wall behind litener (m). C5.; % eflection coefficient for right wall (m). C6.; % eflection coefficient for roof (m). A.; % oudpeaker diaphragm radiu (m). f [::]; % Frequency range. % % Check conitency of input parameter. % if (min([x Y X Y ]) < ) error('coordinate (x,y,z) mut not be negative.'); end if (min([x Y ]) < ) error('oom coordinate mut not be negative.'); end if (min([(x-x) (Y-Y) (-) (X-X) (Y-Y) (-)]) < ) error('oordinate (x,y,z) mut not exceed room limit.'); end if (min([c C C C4 C5 C6]) < ) error('eflection coefficient (C) mut not be negative.'); end if (max([c C C C4 C5 C6]) > ) error('eflection coefficient (C) mut be maximum unity.'); end if (A < ) error('oudpeaker diaphragm diameter mut be poitive.'); end % % Calculate contant vector. Ørted TU Acoutical Technology 98

99 oudpeaker croover network % % itance vector for loudpeaker : P [(X-X) (Y-Y) (-)]; % irect ignal. P [(XX) (Y-Y) (-)]; % eflection. P [(X-X) (YY) (-)]; % eflection. P [(X-X) (Y-Y) ()]; % eflection. P4 [(X-(*X-X)) (Y-Y) (-)]; % eflection 4. P5 [(X-X) (Y-(*Y-Y)) (-)]; % eflection 5. P6 [(X-X) (Y-Y) (-(*-))]; % eflection 6. % oudpeaker direction vector: X co(pi*a/8)*co(pi*av/8); % Coordinate x. Y in(pi*a/8)*co(pi*av/8); % Coordinate y. in(pi*av/8); % Coordinate z. [ X Y ]; [-X Y ]; [ X -Y ]; [ X Y -]; 4 [-X Y ]; 5 [ X -Y ]; 6 [ X Y -]; % % Obervation angle: % T aco(( *P') /(norm() *norm(p))); % irect ignal. T aco((*p')/(norm()*norm(p))); % eflection. T aco((*p')/(norm()*norm(p))); % eflection. T aco((*p')/(norm()*norm(p))); % eflection. T4 aco((4*p4')/(norm(4)*norm(p4))); % eflection 4. T5 aco((5*p5')/(norm(5)*norm(p5))); % eflection 5. T6 aco((6*p6')/(norm(6)*norm(p6))); % eflection 6. % Check that the angl are non-zero (>.e-6) to avoid diviion by zero. if (ab(t)<ep) T ep; end if (ab(t)<ep) T ep; end if (ab(t)<ep) T ep; end if (ab(t)<ep) T ep; end if (ab(t4)<ep) T4 ep; end if (ab(t5)<ep) T5 ep; end if (ab(t6)<ep) T6 ep; end % % irectivity. % k (*pi/4).*f; % xpand k vector. % irectivitie: *beelj(,k*a*in(t))./(k*a*in(t)); % irect ignal. *beelj(,k*a*in(t))./(k*a*in(t)); % eflection. *beelj(,k*a*in(t))./(k*a*in(t)); % eflection. *beelj(,k*a*in(t))./(k*a*in(t)); % eflection. 4 *beelj(,k*a*in(t4))./(k*a*in(t4)); % eflection 4. 5 *beelj(,k*a*in(t5))./(k*a*in(t5)); % eflection 5. 6 *beelj(,k*a*in(t6))./(k*a*in(t6)); % eflection 6. % % Sound preure. % p norm(p)*(/norm(p).*exp(-i*k*norm(p))... (C*/norm(P)).*exp(-i*k*norm(P))... (C*/norm(P)).*exp(-i*k*norm(P))... (C*/norm(P)).*exp(-i*k*norm(P))... (C4*4/norm(P4)).*exp(-i*k*norm(P4))... (C5*5/norm(P5)).*exp(-i*k*norm(P5))... (C6*6/norm(P6)).*exp(-i*k*norm(P6)) ); % % Plot data. Ørted TU Acoutical Technology 99

100 oudpeaker croover network % emilogx(f,.*log(ab(p))); ylabel('sound preure level (db)'); xlabel('frequency (z)'); axi([ - 5]); Ørted TU Acoutical Technology