Linearteam tech paper. The analysis of fourth-order state variable filter and it s application to Linkwitz- Riley filters

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1 Linearteam tech paper The analyi of fourth-order tate variable filter and it application to Linkwitz- iley filter Janne honen 5..

2 TBLE OF CONTENTS. NTOCTON.... FOTH-OE LNWTZ-LEY (L TNSFE FNCTON.... TNSFE FNCTON OF STTE-VBLE CHTECTE.... CCT SYNTHESS EXMPLES ELZTON CCCY EFEENCES...

3 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page of. ntroduction The Linkwitz-iley-filter i conidered to be one of the bet analog croover filter function. t ha many deirable propertie, uch a contant phae between output. eulting ummed tranfer function i of all-pa-type. t exhibit ome phae ditortion, but flat amplitude repone, auming that no delay are introduced after the filter before the ummation point. n thi paper, ll derive neceary equation to deign Linkwitz-iley filter or any other combination of two identical econd-order filter ection by uing tate-variable filter. State variable filter i nice, if you need identical high-pa and low-pa ection imultaneouly, becaue it implement both of them. Furthermore, they have identical cut-off-frequencie, o requirement for preciion component i reduced. Originally, aw thi type of circuit in Finnih book, called akenna HF-laitteita. t decribe thi type of filter, by uing tate-variable filter, but leave reader no clue how the feedback reitor network hould be deigned to have required and. t only tate that the -value i et by deigning feedback loop appropriately. Either the author ha no idea how it i done, or i unwilling to reveal the information. Thi paper hould fill the gap. Furthermore, can t tand circuit which have no clue how the deign i made. My derivation follow guideline given in []. lthough actual circuit here i a bit different, analyi i till baically the ame. There are ome mathematic involved here, but that i a bit unavoidable during analyi of circuit like thi.. Fourth-order Linkwitz-iley (L tranfer function Generalized econd order high-pa tranfer function i defined to be H ( where i the quality factor, i.e. abolute gain at natural angular frequency and i gain at high frequencie, >>. Generally, the fourth order tranfer function i H HP H HP H HP ( Multiplying thee two part together we get for tranfer function

4 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page of H HP ( L tranfer function can be formed by compoing identical two econd order butterworth filter, which have of, ame natural frequency and uually ome deired gain, o we make following aignment: With thee implification, the tranfer function (x i reduced to H HP ( and further to H HP (5. Tranfer function of tate-variable architecture To fit the L tranfer function, we ue baic definition of tranfer function: in H HP (6 To obtain deired form, we ll cro-multiply eq. (7, and divide both ide by : in (7 Preceding equation omewhat give u a hint, how tate-variable filter i formed:

5 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page 5 of Firt, we form high-pa filtered ignal, and applying ucceively integrator, we get to lowpa function after fourth integrator. Problem i, however, how we obtain the high-pa filtered ignal in the firt place. By arranging the equation a in (8 We ee, that ( i obtainable by uing weighted ummer circuit from intermediate output. lo, with thi arrangement, connection to integrator implementation i particularly trong. ealization of thi tranfer function i hown in figure below: Figure. Schematic of fourth order tate-variable filter realization. Thi circuit i mot conveniently analyzed with uperpoition. That mean that we conider only one ource per time, and ret are thought to be zero with zero output impedance. With component reference in figure x, we get following expreion for voltage HP ( at output of the ummer: ( ( ( ( ( ( HP HP HP HP N HP (9 Each integrator time contant i F C F, which i choen to be f C F F π (. Circuit ynthei We jut derived expreion for HP. Now we jut equate equation (x and (y, and get following equation: HP(

6 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page 6 of ( n order to implify thee requirement, we can make following concluion from previou equation: eitor ratio equal one, o we can combine it with ratio o that. That leave u till four unknown ratio and three equation to work with, o we can elect one ratio arbitarily. n previouly mentioned ource, and are equally valued, o we can make ame election. ll mark. Previou et of equation i reduced to

7 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page 7 of ( By making following ubtitution to implify ret of the derivation of ynthei eq:, B, C we can denote above eq a C C B C B ( By combining firt two eq, we can olve reitor ratio, B and C: 6 ( C (5 C B (6

8 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page 8 of 5. Example Example : an example, let deign L filter, what ha following pecification: f c.5 khz and (6. db. Firt we compute contant : 6 ( 6,77 (7 Then C: ( C (8 nd finally B: B C,657 (9 Now when we revert back to original value, we can calculate reitor value:, B, C Chooing a kω, we get and : ko 7,7 ko ( Then, by chooing for and alo a kω, we get : ko 6,6 ko ( B and : ko,5ko ( C

9 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page 9 of C F i choen to be nf, and F calculated: F 57, 8Ω ( 9 πc f π.5 F Finalized circuit chematic look like thi: Figure. Completed L filter realization chematic with (.77,, f c.5 khz. Below i a output from imulation uing previouly calculated reitor value with PSPCE: Figure. Simulated frequency repone of filter in fig..

10 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page of Example : Now, deign again a L Filter, which ha total gain a, and cut-off frequency f c of 5 Hz (ame cae a in []: Firt we compute contant : 6 ( 6,56 ( Then C: ( C (5 nd finally B:,56 B C,6569 (6 Now when we revert back to original value, we can calculate reitor value:, B, C Chooing a kω, we get and : ko,56,56 ko (7 Then, by chooing for and alo a kω, we get : ko 5,kO (8 B,6569 and : ko,5ko (9 C C F i choen to be nf, and F calculated:

11 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page of F,8kO ( 9 πc f π 5 F gain imulation agree well:

12 The analyi of fourth-order tate variable filter and it application to Linkwitz-iley filter Linearteam technical paper Janne honen Page of 6. ealization accuracy State-variable architecture i omewhat enitive to opamp finite gain-bandwith-product. Here i plot of ummed frequency repone from high and lowpa output. Simulation wa here on example, becaue it let you compare with reitor value given in []. deally, it hould be completely flat. ing reitor value from [], there i ome more deviation to repone: pper ummed frequency repone how.8 db deviation from ideal behaviour uing otherwie identical part. Below it, i a trace uing calculated ideal component value. 7. eference [] Sedra-Smith, Microelectronic Circuit [] akenna HF-laitteita, Helinki Media

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