Design of Third-Order Square-Root-Domain Filters Using State-Space Synthesis Method

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1 Deign f Third-Order Square-Rt-Dmain Filter Uing State-Space Synthei Methd Ali Kircay 1, M. Serhat Keerliglu, F. Zuhal Sagi 1 1 Harran Univerity, Electrical and Electrnic Engineering, 63 Sanliurfa, Turkey kircay@harran.edu.tr; fatmazuhalagi@harran.edu.tr 1 Pamukkale Univerity, Electrical and Electrnic Engineering, 7 Denizli, Turkey mkeerliglu@pau.edu.tr Abtract n thi tudy, quare-rt-dmain electrnically tunable, third rder lw-pa filter i prped. Firt circuit i thirdrder lw-pa Butterwrth filter and ecnd circuit i thirdrder lw-pa hebyhev filter. Additinally, the cut-ff frequency f the prped filter can be electrnically tuned by changing external current. Time and frequency dmain imulatin are perfrmed uing PSPE prgram fr third rder filter t verify the thery and t hw the perfrmance f them. Fr thi purpe, the filter are imulated by uing TSM.35 μm Level 3 MOS prce parameter. 1. ntrductin Square-rt-dmain circuit are a ubcla f cmpanding circuit prviding lw pwer under lw-vltage, having large dynamic range, perating in high frequencie and electrnically tunability uing D current urce. Due t thee prpertie, cmpanding circuit are cmpatible with MOS very large cale integratin technlgy. Beide thee prpertie, circuit are implemented in thi technlgy; deign f cmpanding circuit ha received great attentin. Lg-dmain and quare-rtdmain filter circuit are an applicatin f cmpanding methd. Thee circuit can be aid that the mt widely ued tranlinear circuit. Lg-dmain circuit are prped by Adam [1] and then they have been tudied by Frey [, 3]. Baic tranlinear principle ue the expnential -V characteritic f BJT r MOSFET in weak inverin regin [4, 5]. MOS tranlinear principle (MTL) derived frm biplar tranlinear principle (BTL) [4] by Seevinck [6] ue quadratic relatinhip between vltage and current f MOS tranitr in trng inverin and aturatin regin. Starting frm tate-pace equatin, a well a quadratic relatinhip between the vltage and current f the MOS tranitr, filter perfrmed by uing analg prceing circuit blck uch a quare-rt and quarer/divider circuit are called quare-rt-dmain filter [7 3]. Square-rt-dmain firt-rder filter circuit [14, 16, 18, 1], ecnd rder vltage-mde [11, 1, 13] r current-mde [9, 14,, 3] filter circuit and trancnductance and tranreitance circuit [16, 17, 4] have been tudied by variu reearcher. Hwever, a a reult f the literature urvey, it ha been een that the tudie n quare-rt-dmain third-rder filter circuit wa fund t be minimal. Third-rder filter circuit btained uing OTA and OTRA are preented in [5-7]. A tate pace la AB ynthei methd fr the deign f quare-rtdmain filter baed n the MOSFET quare law i prped in [8]. A third-rder lw-pa elliptic filter uing a quare-rtdmain differentiatr i preented in [9]. A third-rder elliptic lw-pa L filter i prped in [3]. A realizatin f thirdrder active witched-capacitr filter i prped in [31]. n thi tudy, quare-rt-dmain third rder lw-pa Butterwrth and hebyhev filter have been deigned uing tate-pace-ynthei methd with quare-rt and quarer/divider circuit, MOS current mirrr, D current urce, D upply vltage, and grunded capacitr. Statepace ynthei methd i a very ueful and efficient apprach fr the deign f cmpanding circuit [3]. t prvide a general lutin fr realizing circuit functin. The key apect f the ue f tate-pace methd in thi tudy i that exactly relate internally nn-linear filter t equivalent linear ytem. ut-ff frequency f the prped filter can be adjuted electrnically by changing the value f the D current urce.. Square-Rt-Dmain Third-Order Filter Deign A general third-rder circuit functin i a hwn in Equatin (1). P () () P() N = = (1) ( ) ω1 ω1 ( ω ) Equatin (1) can be tranfrmed t the fllwing tatepace equatin. ω x 1 1 = x1 ωx3 () = ω x u (3) x ω (4) x3 = ω 1x1 ω1x Here, x 1, x and x 3 repreent the tate variable. They are gate-urce vltage f MOS tranitr. Hence, the fllwing tranfrmatin can be applied t the quantitie in the equatin [18, 19]. i = ( Vi Vth ), i = 1,,3 (5) n Equatin (5), i repreent drain current f MOS tranitr in aturatin regin, =μ x (W/L) tand fr trancnductance, V i repreent gate-urce vltage and V th repreent the threhld vltage. V i vltage can be btained a hwn in the fllwing equatin: V = i i Vth, i = 1,,3 (6) 916

2 The relatin given abve can be rganized t yield the ndal equatin belw after they are applied t Equatin (), (3) and (4). V 1 ω1 ω = 3 1 ω1v th (7) ω ω V = u (8) ω1 ω V 1 3 = 1 (9) n thee equatin, i a capacitr value reembling a multifunctin factr. V, 1 V and V 3 in Equatin (7), (8) and (9) can be accepted a time dependent current that are grunded via three capacitr. The fllwing equatin 1, and B can be defined fr ue in Equatin (7), (8) and (9). ω 1 1 = (1) ω = (11) 1 1 B V th = 1 Equatin (7), (8) and (9) can be arranged a fllw: (1) V = B (13) V u = (14) 11 1 V 3 = (15) f we accept the quality factr a =1 and divide bth ide f the equatin by. The tate equatin in Equatin (13), (14) and (15) can be written a belw: 11 1 V 3 1 = (16) V u = (17) 11 1 V 3 = (18) The fllwing definitin applie t Equatin (16), (17) and (18). = (19) Square-rt-dmain third rder vltage-mde lw-pa Butterwrth filter circuit i hwn in Figure 1 ha been actualized uing Equatin (16), (17) and (18). U, V 1, V and V 3 repreent the input and utput vltage f the filter circuit, repectively. Additinally, uing Equatin () and (1), ω 1 and ω ple frequency f the filter circuit can be determined depending n the 1,, and [9, 18]. 1 ω 1 = () ω = (1) Uing Equatin (), (3) and (4), utput variable f the quare-rt-dmain third-rder lw-pa filter circuit can be determined depending n ω 1, ω and. ω V 1ω 1 = U () ω 1 ω ( ) 1 ω ω 1 ω ω1ω V = U (3) ω 1 ω ( ) 1 ω ω 1 ω1ω V3 = U (4) ω 1 ω ( ) 1 ω n accrdance with Equatin (), the utput f the circuit a hwn in Figure 1 prvide a third rder inverting lw-pa filter tranfer functin. V AG = V 1 (5) nequently, uing the utput f the circuit hwn in Figure 1 fr =1 and ω 1 =ω =ω, third-rder Butterwrth lwpa filter vltage tranfer functin accmplihed a defined in Equatin (6). 3 ω V 1 = ( )( ) U (6) ω ω ω f we accept the quality factr a 1 in Equatin (13), (14) and (15), quare-rt-dmain third rder lw-pa hebyhev filter circuit can be perfrmed a hwn in Figure. n accrdance with Equatin (), the utput f the circuit a hwn in Figure prvide a third rder inverting lw-pa hebyhev filter tranfer functin. n cae f 1 in Equatin (), the equatin between 1 and given by Equatin (7) mut be atified. = 1 (7) With 1 db paband ripple and ω =1rad/ cut-ff frequency, third rder nrmalized lw-pa hebyhev filter ω 1, ω and value given in Equatin () are a hwn belw [3]. ω 1 =,997, ω =, 494, =,

3 Fig. 1. Square-rt-dmain third rder lw-pa Butterwrth filter circuit Fig.. Square-rt-dmain third rder lw-pa hebyhev filter circuit 3. Simulatin Reult TSM.35 µm Level 3 MOS tranitr parameter [33] have been ued in PSPE imulatin f the deigned quarert-dmain third rder vltage-mde lw-pa Butterwrth and hebyhev filter. Tranitr dimenin are chen a W/L=1 μm/1 μm fr M 1 ~M 6, W/L= μm/ μm fr M 7 ~M 14. The circuit upply vltage i elected t be V DD =3 V. The value f three capacitance f the circuit are chen t be = pf. The imulatin are perfrmed t tune the cut-ff frequency by varying the value f the current urce. Varying the value f the current urce frm 1 µa t 4 µa, the cutff frequency f the filter i tuned frm 157 khz t 857 khz. A a reult, the cut-ff frequency f the filter can be adjuted in the 7 khz frequency range. The gain repne btained fr the different value f the D current urce f the third rder lwpa Butterwrth filter circuit have been given in Figure = 1uA = 81uA = 4uA 1E4 1E5 1E6 1E7 Fig. 3. Electrnically tunable prpertie f Butterwrth filter The phae repne btained fr the different value f the D current urce f the third rder lw-pa Butterwrth filter circuit have been given in Figure

4 1 Phae Angle [Degree] = 1uA = 81uA = 4uA = 1uA 1 = 88uA -6 1E4 1E5 1E6 1E7 Fig.4. Electrnically tunable phae repne f Butterwrth filter The time-dmain repne f the Butterwrth filter i hwn in Figure 5..5 V ine-wave input at a frequency f 85 khz wa applied t the filter. The ttal harmnic ditrtin wa meaured a 1.6%. -6 1E4 1E5 1E6 1E7 Fig.6. Electrnically tunable prpertie f hebyhev filter The gain repne f hebyhev filter btained fr 1 =88 μa with 1 db paband ripple i hwn a larger in Figure = 4uA Amplitute [V] Vin Vut 1.E-4 1.1E-4 1.E-4 1.3E-4 1.4E-4 1.5E-4 Fig.5. Time dmain repne f the prped Butterwrth filter The gain repne btained fr the different value f the D current urce f the third rder lw-pa hebyhev filter circuit have been given in Figure 6. D current urce value are B =4.65 μa and =.79 μa, =1.6 μa and =98.7 μa fr 1 =1 μa, 1 =88 μa and 1 =4 μa, repectively. Al, the value f D input current f the blck i cnnected t the M 5 tranitr are 1 =4.11 μa, 1 =35.9 μa and 1 =195 μa, repectively. -5 1E4 1E5 1E6 Fig.7. hange f hebyhev filter paband The time-dmain repne f the hebyhev filter i hwn in Figure 8.. V ine-wave input at a frequency f 8 khz wa applied t the filter. The ttal harmnic ditrtin wa meaured a 1.5%...1 Amplitute [V]. 1.9 Vin Vut E-4 1.1E-4 1.E-4 1.3E-4 1.4E-4 1.5E-4 Time [] Fig. 8. Time dmain repne f the prped hebyhev filter 919

5 4. ncluin Square-rt-dmain third rder vltage-mde lw-pa Butterwrth and hebyhev filter circuit are prped in thi tudy. A ytematic ynthei prcedure t derive the filter circuit i al given. Thee circuit cnit f nly MOS tranitr and grunded capacitr. ut-ff frequency f the filter circuit can be adjuted electrnically by changing the value f the D current urce. The mt imprtant feature f the circuit i electrnic tunability, that i, the gain and phae repne f the circuit can be cntrlled by D current urce. PSPE imulatin are prvided t cnfirm the theretical analyi. 5. Reference [1] Adam, R. W., Filtering in the lg-dmain, in 63 rd AES nf., New Yrk, 1979, preprint 147. [] Frey, D. R., Lg-dmain filtering: An apprach t currentmde filtering, EE Prc. G, vl. 14, pp , [3] Frey, D. R., Expnential tate-pace filter: A generic current mde deign trategy, EEE Tran. ircuit Syt., vl. 43, pp. 34-4, Jan [4] Gilbert, B., Tranlinear circuit: A prped claificatin, Electrnic Letter, vl. 11(1), pp , Jan [5] Ngarmnil, J., and Tumazu,., Micrpwer lg-dmain active inductr, Electrnic Letter, vl. 3(11), pp , [6] Seevinck, E., and Wiegerink, R. J., Generalized tranlinear circuit principle, EEE J. Slid-State ircuit, vl. 6(8), pp , [7] Payne, A., and Tumazu,., Linear tranfer functin ynthei uing nn-linear cmpnent Prceeding SAS'96, Atlanta, USA,, pp [8] Ekiyerli, M. H., Payne, A.J., and Tumazu,., State- Space Synthei f ntegratr Baed n The MOSFET Square Law, Electrnic Letter, vl. 3, n. 6, [9] Ekiyerli, M. H., and Payne, A. J., Square Rt Dmain filter deign and perfrmance, Analg ntegrated ircuit Signal Prceing,, pp ,. [1] Lpez-Martin, A. J., and arlena, A., Sytematic deign f cmpanding ytem by cmpnent ubtitutin, Analg ntegrated ircuit Signal Prceing, 8, pp , 1. [11] Yu, G.J., Liu, B.D., Hu, Y.., and Huang,.Y., Deign f Square-Rt Dmain Filter, Analg ntegrated ircuit and Signal Prceing, 43, 49 59, 5. [1] Menekay, S., Tarcan, R.., and Kuntman, H., The Secndrder lw-pa filter deign with a nvel higher preciin quare-rt circuit, tanbul Univ., J. Electr. Electrn., 7, 1, 33 79, 7. [13] Ekiyerli, M. H., Payne,A.J., and Tumazu,., State- Space Synthei f Biquad Baed n The MOSFET Square Law, ircuit and Sytem, SAS '96, vl. 1.pp , Atlanta, GA, USA, 1 15 May [14] Kumar, J. V., and Ra, K. R., A Lw-Vltage Lw Pwer MOS mpanding Filter, Prceeding f the 16th nternatinal nference n VLS Deign (VLS 3) /3, 3. [15] Mulder, J., Static and dynamic tranlinear circuit, Delft Univerity Pre, Netherland, [16] Lpez-Martin,A. J., and arlena, A., Very lw vltage MOS cmpanding filter baed n the MOS tranlinear principle, Mixed-Signal-Deign, 1, SSMSD, Suthwet Sympium n, pp [17] Ragheb, T. S. A., and Sliman, A. M., New Square-Rt Dmain Ocillatr, Analg ntegrated ircuit and Signal Prceing, 47, , 6, DO: 1.17/ [18] Lpez-Martin, A. J., and arlena, A., 1.5 V MOS cmpanding filter, Electrnic Letter, Vl.: 38, ue:, pp.: ,. [19] Mulder, J., Serdijn, W. A., Van Der Werd, A.., and Van Rermund, A. H. M., A 3.3 Vlt urrent-ntrlled - Dmain Ocillatr, Analg ntegrated ircuit and Signal Prceing, 16, 17-8, [] Menekay, S., Tarcan, R.., and Kuntman, H., Nvel highpreciin current-mde-circuit baed n the MOStranlinear principle nt. J. Electrn. mmun. (AEU) DO:1.116/j.aeue.8.8.1, 8. [1] Keerliglu, M. S., and Kircay, A., The deign f currentmde electrnically tunable firt-rder quare-rt dmain filter uing tate-pace ynthei methd, nternatinal Review n Mdelling and Simulatin, Vl., N., pp.14-18, April 9. [] Kircay, A., and Keerliglu, M. S., Nvel urrent-mde Secnd-Order Square-Rt-Dmain Highpa and Allpa Filter, 6th nt. nf. n Elecrical and Electrnic Engineering, 5-8 Nvember 9, Bura, Türkiye [3] Kircay, A., Keerliglu, M. S., and am, U., urrent- Mde Square-Rt-Dmain Ntch Filter Eurpen nference n ircuit Thery and Deign 9, Antalya, Augut 3 7, 9 [4] Keerliğlu, M. S., Square-Rt-Dmain Firt Order Tranadmittance and Tranimpedance Type Filter Deign ELEO 1, Elektrik-Elektrnik-Bilgiayar Mühendiliği Sempzyum ve Fuarı, Bura, Turkey, 5 December 1. [5] mail S. H., Sliman E. A., and Mahmud S. A., acaded Third-rder Tunable Lw-Pa Filter uing Lw Vltage Lw Pwer OTA, nternatinal Sympium n ntegrated ircuit, , 11. [6] Ghh M., Paul S. K., Ranjan R. K., and Ranjan A., Third Order Univeral Filter Uing Single Operatinal Tranreitance Amplifier, Hindawi Publihing rpratin Jurnal f Engineering, Article D 31796, 6 page, 13. [7] Sun Y., Jefferıe B., and Teng J., Univeral third-rder OTA- Filter, nt. J. Electrnic, vl. 85, n. 5, , [8] Surav Yilmaz S., and Tla A. T., A ytematic cla AB tate pace ynthei methd baed n MOSFET quare law and tranlinear quare-rt cell, nt. J. irc. Ther. Appl., DO: 1.1/cta.54, 15. [9] Fuad K.O.M., and Sliman A.M., Square rt dmain differentiatr, EE Prc.-ircuit Device Syt., Vl. 15, N. 6, December 5. [3] Pychalin., Square-Rt Dmain Operatinal Simulatin f L Ladder Elliptic Filter, ircuit Sytem Signal Prceing, vl. 6, n., pp. 63 8, 7. [31] Shinde1 G. N., Bhagat S. R., Tunable Bandwidth Third Order Switched-apacitr with Multiple Feedback Filter fr Different enter Frequencie, Science Reearch Engineering,, , 1. [3] Kendall, S., Analg Filter nd ed., Kluwer Academic Publiher,. [33] Minaei, S., Dual-nput urrent-mde ntegratr and Differentiatr Uing Single DV and Grunded Paive Element EEE MELEON 4, May 1 15, Dubrvnik, ratia. 9

Current/voltage-mode third order quadrature oscillator employing two multiple outputs CCIIs and grounded capacitors

Current/voltage-mode third order quadrature oscillator employing two multiple outputs CCIIs and grounded capacitors Indian Jurnal f Pure & Applied Physics Vl. 49 July 20 pp. 494-498 Current/vltage-mde third rder quadrature scillatr emplying tw multiple utputs CCIIs and grunded capacitrs Jiun-Wei Hrng Department f Electrnic