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