Jounal of icuits, Systems, and omputes Vol. 7, No. (2008) 33 54 c Wold Scientific Publishing ompany HISTOY ND POGESS OF THE TOW THOMS BIQUDTI FILTE. PT I. GENETION ND OP MP ELIZTIONS HMED M. SOLIMN Electonics and ommunication Engineeing Depatment, aio Univesity, Egypt 263 asoliman@ieee.og evised 0 pil 2007 The histoy of Tow Thomas secondode filte is eviewed. Two altenative geneation methods of the Tow Thomas filte ae discussed. The fist is a geneation method fom the secondode passive L filte and the second is fom the multiple feedbacks inveting lowpass filte using a single op amp. Seveal foms of the cicuit ae biefly eviewed. Passive and active compensation methods to impove the cicuit pefomance fo highq designs ae summaized. Spice simulation esults ae included. Keywods: ctive filtes; opeational amplifie cicuits; Tow Thomas filte.. Intoduction One of the most famous active filte cicuits is the Tow Thomas biquadatic cicuit (TTbiquad).,2 This cicuit is included almost in all textbooks in active filtes, 3 0 and is intoduced in most univesities to the undegaduate o gaduate students. lthough the cicuit was intoduced since 38 yeas it is still eceiving inteest of eseaches in modifying it to fit the new MOS technology. 3 Due to the geat impotance of this cicuit and the pogess in its ealization it is desiable to collect such pogess in a eview pape, this is the objective of this pape. It is hoped that this pape will be of value to pofessos teaching this cicuit and to designes of active filtes. 2. The Oigin of the Tow Thomas Filte icuit lthough the cicuit is well known by the names of the two pesons it is impotant to know the oiginal histoy of the cicuit. The cicuit was fist intoduced by Tow. In his pape, he intoduced five cicuits to ealize diffeent filte esponses lowpass, bandpass using thee op amps and then highpass, allpass and notch using fou op amps. Thomas independently submitted his pape 2 in July 969 befoe Tow cicuit was published in Decembe 969; that is why the cicuit is known by both names. 33
34. M. Soliman Thomas 2 stated his pape with the biquad cicuit in its geneal fom using fou op amps. The pape discussed the Q enhancement caused by finite gain bandwidth of the op amp and it was among the fist papes 4 discussed this vey impotant topic. The TT cicuit can be geneated fom passive L filte o fom the multiple feedbacks inveting lowpass filte as will be shown next. 2.. Geneation of TT cicuit fom passive L cicuit Figue (a) epesents the passive L filte, ealizing a lowpass output esponse at the capacito teminal. The tansfe function is given by = s 2 L s. () This tansfe function can be epesented by the block diagam shown in Fig. (b). 5 The block diagam can be epesented by two cascaded lossy integato and lossless integato as shown in Fig. (c), whee fo the passive L cicuit in Fig. (a), the time constants τ and τ 2 ae given by τ = L, τ 2 =. (2) L (a) V I Σ S 2 L S (b) Σ sτ τ s 2 (c) Fig.. (a) The passive L lowpass filte. (b) block diagam of (a). (c) n equivalent block diagam to that of (b).
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 35 a b Σ sτ sτ2 (a) 3 4 2 2 (b) Fig. 2. (a) modified block diagam to that of Fig. (c). (b) The Tow Thomas cicuit.,2 Next, some modifications ae intoduced to this block diagam by adding the paametes a, b to povide degees of feedom in contolling the gain and the Q facto of the active filte. The modified block diagam is shown in Fig. 2(a) and its tansfe functions ae given by = = ab τ s s 2 τ s ab τ τ 2 s 2 τ s a, (3a) τ τ 2 a. (3b) τ τ 2 The block diagam in Fig. 2(a) is pactically ealizable to give the wellknown TT biquad shown in Fig. 2(b), dawn hee with the and suffixes as in Thomas pape exactly. 2 Fom Figs. 2(a) and 2(b) it can be seen that: τ =, τ 2 = 2 2, a =, b = 3. (4) 3 4 The bandpass and the lowpass tansfe functions ae given, espectively, by = = s 2 s 2 s 4 s, (5a) 2 2 3 2 2 4 s 2 2 3. (5b)
36. M. Soliman Fom the above equations it is seen that the ω 0 and Q ae given by ω 0 =, Q =. (6) 2 2 3 2 2 3 The magnitude of the gain at ω 0 at the bandpass output is given by T (ω 0 )= / 4. The magnitude of D gain at both of the lowpass outputs is given by T (0) = 3 / 4. Taking = 2 =, 2 = 3 =, the design equations fo a specified ω 0 and Q ae given by = Q, = ω 0. (7) Fo a specified bandpass cente fequency gain T (ω 0 ), the design equation fo 4 is given by 4 = T (ω 0 ). (8) Fo a specified D gain T (0), the design equation fo 4 is given by 4 = T (0). (9) 2.2. Geneation of TT cicuit fom a single op amp lowpass filte Figue 3(a) epesents the single op amp multiple feedback inveting lowpass filte. 3 pplying Kichoff cuent law (KL) at the bandpass node esults in: Let G 4 G 3 = (G 2 G 3 G 4 s ). (0) G = G 2 G 3 G 4. () Equation (0) can be ealized by the fist two stages of Fig. 3(b) (in which is negative) which when combined with the second inveting integato esults in a vesion of the TT cicuit shown in Fig. 3(b). The esisto can now be taken abitay and diffeent fom the value given by Eq. (). The design equations ae the same as given by Eq. (7). The cicuit in Fig. 3(b) has two bandpass outputs one inveting and one noninveting and has only one inveting lowpass output. That is the main diffeence fom the TT cicuit shown in Fig. 2(b). 2.3. Modified foms of Tow Thomas cicuit n altenative fom of TT cicuit was also intoduced in ef. 6 by adding the esisto 5 as shown in the cicuit in Fig. 3(c). The cicuit has one inveting
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 37 3 4 2 2 (a) 3 2 4 2 (b) 3 4 2 2 5 (c) Fig. 3. (a) The single op amp multiple feedback lowpass filte. 3 (b) ltenative fom of Tow Thomas cicuit. 6 (c) modified fom of the cicuit in (b). 6 (d) Modified Tow Thomas cicuit using Deboo integato. 2,4
38. M. Soliman 3 4 2 5 2 2 (d) Fig. 3. (ontinued ) lowpass output and two opposite polaity bandpass outputs. The cicuit equations ae given by = = s 2 s s 2 s ( ( s 4 5 ) 2 2 4 5 ) 2 2 3, (2a) 2 2 3. (2b) It is seen that the esistos and 5 contol Q of the filte which can be made high when the two esistos appoach each othe in value. The subtaction, howeve, in the s tem esults in a highq sensitivity to and 5 and is given by Q S = Q S 5 = Q ω 0. (3) It should be noted that the oiginal TT cicuit shown in Fig. 2(b) has all the ω 0 and Q passive sensitivities to all passive cicuit components being less than o equal to. modified fom of the Tow Thomas cicuit based on Deboo gounded capacito noninveting integato is shown in Fig. 3(d). 2,4 The cicuit uses one op amp less than the oiginal Tow Thomas cicuit and ealizes inveting bandpass and inveting lowpass esponses as shown in Table. necessay condition fo the noninveting integato is that 5 = 2. The cicuit has high passive Q sensitivity to 5 and 2 and fo independent contol on Q these two esistos must be equal. In ef. 4 the cicuit was used without the feedback esisto and in this case thee is no independent contol on Q.
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 39 Table. ompaison of the Tow Thomas biquad and the modified cicuits. icuit Independent Bandpass Lowpass figue Op amps contol on Q polaity polaity 2b 3 2 6 Yes by Inveting Both polaities 3b 3 2 6 Yes by Both polaities Inveting 3c 3 2 7 Yes by Both polaities Inveting o 5 3d 2 2 7 Yes by Inveting Inveting 4a 4 2 9 No Both polaities 4b 3 2 7 No Both polaities 4c 3 2 7 No Both polaities 4d 3 2 6 Yes by Both polaities 3. Modified TT icuits with LowPass Output Only In this section, thee altenative foms of the cicuit shown in Fig. 3(c) and ealizing only lowpass esponse ae summaized. The thee cicuits ae shown in Figs. 4(a) 4(c) and they ae equivalent to each othe. 6 The tansfe function of the inveting lowpass is given by = ( s 2 s 2 2 4 ) ( 2 5 2 2 3 5 ). (4) 3 2 4 2 5 (a) Fig. 4. (a) modified fom of the cicuit in Fig. 3(b). 6 (b) simplified cicuit in (a). 6 (c) ltenative equivalent cicuit to (a) and (b). 6 (d) ltenative fom of Tow Thomas cicuit ealizing lowpass only. 7
40. M. Soliman 3 4 2 2 5 (b) 3 4 2 5 2 (c) 3 4 2 2 (d) Fig. 4. (ontinued ) nothe altenative fom of TT cicuit obtained by intechanging the positions of the lossless and the lossy integatos in the feedback loop of the cicuit in Fig. 2(b) is shown in Fig. 4(d). 7 This cicuit ealizes only lowpass esponses with both polaities. Its tansfe function is given by = s 2 2 2 4 s 2 2 2 3. (5)
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 4 ompaed with the thee equivalent lowpass filtes descibed by Eq. (4), it can be seen that this cicuit has independent contol on Q by vaying which cannot be achieved with any of the thee equivalent cicuits in Figs. 4(a) 4(c). 4. Fequency Limitation Equations In efs. 6 and 7 the singlepole model of the op amp is taken into consideation and the op amp gain is epesented by (s) = ω t s, (6) ω t is the gain bandwidth poduct of the op amp. It was poved in ef. 6 that the actual ω 0 and Q fo the cicuit shown in Fig. 4(d) ae given as ω 0 = ω 0, (7) ω 0 ω t Q Q = 4ω 0Q ω t. (8) The above equations indicate that the deviation in ω 0 is vey limited wheeas the majo deviation is in Q. Equation (8) can be used to find the maximum fequency of opeation of the cicuit fo a specified op amp, ω 0 and Q and the allowable Q. The above equations apply also to the cicuits in Figs. 2(b) and 3(b). Thus, it is seen that it is desiable to have phase compensation fo the Tow Thomas cicuit and its modified vesions. Thomas has shown that 2 the total phase lag aound the loop is elated to the Q facto by φ = = Q a Q 4ω 0, (9) ω t Q a is the actual quality facto and Q is the ideal pole Q. The above esult is exactly the same as given by Eq. (8). 5. ompensation of the TT icuit Thee ae two types of compensation, passive compensation and active compensation. Befoe discussing these two types of compensation it is woth to mention the selfcompensated TT cicuit epoted in ef. 8 which educes the total phase lag aound the loop to (2ω 0 /ω t ) which is half the value given by Eq. (9) by just changing one cicuit connection as shown in Fig. 5(a). 8 5.. Passive compensation Thomas 2 added a compensating capacito in paallel with one of the loop esistances to poduce an amount of phase lead equal to 4ω 0 /ω t so that phase coection aound
42. M. Soliman 3 4 2 2 (a) 3 4 2 c 5 2 2 (b) 3 VI 4 c 2 2 (c) Fig. 5. (a) Patial compensation of the TT cicuit using feedfowad technique. 8 (b) Passive compensation applied to the cicuit in Fig. 3(d). 9 (c) Passive compensation applied to the cicuit in Fig. 4(d). 9 the loop is achieved. Othe foms of passive compensation wee given in ef. 9 based on adding a single compensating esisto with one of the two capacitos to ealize a phaselead integato. djusting the phase lead of the integato will esult in a pefect compensation fo the unwanted phase shift poduced by the op amp
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 43 pole. Figue 5(b) epesents one of the cicuits given in ef. 9 whee c epesents the compensating esisto added in seies with the capacito 2 2 of the Deboo integato. The design value fo c is given by 3/(2ω t ). Figue 5(c) epesents anothe fom of passive compensation given in ef. 9, whee c epesents the compensating esisto added in seies with the capacito of the inveting lossless integato. Fo phase compensation the design value fo c is given by 4/(ω t ). 5.2. ctive compensation Seveal methods of active compensation of the two integato loop filtes will be summaized in this section. The fist method is based on eplacing the lossless integato in the cicuit in Fig. 2(b) by a phaselead inveting integato as shown in Fig. 6(a). 20 The excess phase of this integato is given by φ Kω ( ) 0 a ω t K. (20) Thus, it is seen that the esisto /a contols the phase of the integato which can be made leading. The esisto K contols the integato stability. Taking the second op amp pole into consideation it is seen that taking K = will ensue the integato stability. 20 In this case a istakenequalto8topovideaphaseleadof 3ω 0 /ω t which is the amount necessay fo phase coection at ω 0. /a /b Q (K) K VLP (a) Fig. 6. ctive compensated Tow Thomas cicuit using phaselead (a) inveting integato, 20 (b) inveting amplifie, 2 and (c) Deboo integato. 22 (d) The kebeg Mossbeg active compensated TT cicuit. 23 25 The active compensated TT cicuit using phaselead (e) inveting amplifie 26 and (f) inveting amplifie. 27
44. M. Soliman Q K VI /b VLP K /a (b) Q /b 2 (c) Fig. 6. (ontinued ) /a n altenative active compensated cicuit is given in Fig. 6(b) which epesents anothe active compensated vesion of Tow Thomas cicuit suitable fo high Q and high fequencies and using a phaselead inveting amplifie to eplace the unity gain invete stage. 2 To povide the amount of phase lead necessay fo phase coection aound the loop, a is taken as a = K K. (2)
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 45 /b Q (d) Q K 2 /b K (e) Q K 2 /b K (f) Fig. 6. (ontinued) Taking K = 2, esults in an unconditionally stable inveting amplifie, 2 in this case a =2.5. The active compensated cicuit in Fig. 6(c) is based on using a phaselead Deboo integato 22 adjusted to have a phase lead equal to (ω 0 /ω t ), in this case the paamete a in the cicuit in Fig. 6(c) is taken equal to 4. wellknown active compensated TT cicuit is shown in Fig. 6(d) which was intoduced in ef. 23 and known by the kebeg Mossbeg cicuit and it is included in some textbooks also. 23 25 In ef. 25, howeve, the second op amp polaity should be evesed. Two
46. M. Soliman equivalent active compensated TT cicuits using phaselead inveting amplifies ae given in Figs. 6(e) and 6(f). 26,27 The phase of the inveting amplifie at ω 0 is given (K 2 K )ω 0 /ω t,thusfok =,K 2 istakenequalto3topovidethenecessay phase lead of 2ω 0 /ω t. nothe cicuit based on applying active phase compensation sepaately to the Mille integato and to the Deboo integato given in ef. 22 which uses an additional op amp in the feedback loop of the Mille integato. 28 nothe vaiablephase inveting amplifie that has been used fo phase coection in the Tow Thomas cicuit is given in ef. 29. 6. Spice Simulations In this section, Spice simulations ae caied out using the op amp µ 74 fom nalog Devices with f t = MHz and biased with supply voltages of ± 5 V. The TT cicuit in Fig. 2(b) is designed fo Q =0andf 0 =5.92 khz taking = 2 = 500 pf, 2 = 3 = 4 = =20kΩand = 200 kω. Figue 7(a) epesents the Spice simulation esults of the magnitude and phase of the TT cicuit in Fig. 2(b) togethe with the ideal esponse. It is seen that the Q has inceased dastically fom its theoetical value of 0. Fom Fig. 7(a) Spice simulation esults of the Tow Thomas cicuit in Fig. 2(b) simulation esults Q/Q =0.73 which is slightly highe than the theoetical expected value obtained fom Eq. (8) as (a) Fig. 7. Spice simulation esults of (a) the Tow Thomas cicuit in Fig. 2(b), (b) the cicuit in Fig. 3(b), (c) the cicuit in Fig. 3(d), and (d) the cicuit in Fig. 4(d).
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 47 (b) (c) Fig. 7. (ontinued ) 0.64. Figue 7(b) epesents the Spice simulation esults of the magnitude and phase of the TT cicuit in Fig. 3(b) designed with the same values as the pevious cicuit togethe with the ideal esponse. It is seen that the Q/Q isthesameasin the cicuit in Fig. 2(b). Figue 7(c) epesents the Spice simulation esults of the
48. M. Soliman (d) Fig. 7. (ontinued ) magnitude and phase of the TT cicuit in Fig. 3(d) designed with = 2 = 500 pf, 2 = 3 = 4 = 5 = =20kΩand = 200 kω. It is seen that the Q/Q =0.3 which is lowe than the eo in the pevious two cases and is vey close to the theoetical expected value given by 3ω 0 Q/ω t =0.48. The TT cicuit in Fig. 4(d) is designed to ealize a maximally flat magnitude lowpass esponse having Q = 0.707 and f 0 = 59.2kHz taking = 2 =50pF, 2 = 3 = 4 = =20kΩand =4.4 kω. Figue 7(d) epesents the Spice simulated magnitude and phase esponses togethe with the ideal esponses. It is seen that thee is a peak in the magnitude esponse due to the Q enhancement effect. Figue 8(a) epesents the Spice simulated magnitude and phase esponses togethe with the ideal esponses of the patial compensated cicuit in Fig. 5(a). Fom the simulation esults Q/Q = 0.25 which is vey close to the theoetical value given by 2ω 0 Q/ω t =0.32. Figue 8(b) epesents the Spice simulated magnitude and phase esponses togethe with the ideal esponses of the passive compensated cicuit in Fig. 5(b) based on the addition of the compensating esisto c which is taken as 350 Ω to povide pefect compensation. It should be noted that the value taken by c is slightly lowe than the theoetically calculated value of 3/(2ω t ) = 477 Ω. The passive compensated TT lowpass filte in Fig. 5(c) is designed to ealize a maximally flat magnitude lowpass esponse taking Q = 0.707 and f 0 = 59.2 khz taking all cicuit components as befoe with c added and taken as 4/(ω t )=2.74 kω. Figue 8(c) epesents the magnitude and phase simulation
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 49 (a) (b) Fig. 8. Spice simulation esults of (a) the patialcompensated TT cicuit in Fig. 5(a), (b) the passivecompensated TT cicuit in Fig. 5(b), and (c) the passivecompensated TT cicuit in Fig. 5(c).
50. M. Soliman (c) Fig. 8. (ontinued ) esults togethe with the ideal esponses. It is seen that the esponse is vey close to the ideal one indicating a pefect passive compensation. Next, the simulation esults fo thee of the activecompensated cicuits designed fo Q =0andf 0 =5.92 khz and taking b =, = 500 pf, =20kΩaegiven. Figue 9(a) epesents the Spice simulated magnitude and phase esponses togethe with the ideal esponses of the activecompensated cicuit in Fig. 6(a), the esisto (K ) is taken as 40 kω and the esisto (/a) is taken as 2.5kΩ. Figue 9(b) epesents the Spice simulated magnitude and phase esponses togethe with the ideal esponses of the activecompensated cicuit in Fig. 6(b), taking K = 2, in this case and fo phase compensation a =2.5. The esisto /a is taken as 8.5 kω which is slightly highe than its calculated value of 8 kω fo pefect compensation. Figue 9(c) epesents the Spice simulated magnitude and phase esponses togethe with the ideal esponses of the cicuit in Fig. 6(d). The activecompensation simulation esults in all of the povided thee simulations agee well with the ideal esponses as seen fom Figs. 9(a) 9(c). 7. onclusions The histoy of Tow Thomas secondode filte is eviewed. Two altenative geneation methods of the Tow Thomas filte ae discussed. The fist is a geneation method fom the secondode passive L filte and the second is fom the multiple
Histoy and Pogess of the Tow Thomas BiQuadatic Filte. Pat I 5 (a) (b) Fig. 9. Spice simulation esults of (a) the activecompensated Tow Thomas cicuit in Fig. 6(a), (b) the activecompensated Tow Thomas cicuit in Fig. 6(b), and (c) the kebeg Mossbeg cicuit in Fig. 6(d).
52. M. Soliman (c) Fig. 9. (ontinued ) feedbacks inveting lowpass filte using a single op amp. Seveal foms of the cicuit ae biefly eviewed. Table summaizes the polaities of the ealizable outputs and the numbe of cicuit components in each of the cicuits. The oiginal Tow Thomas cicuit has independent contol on Q, some of the modified cicuits loose this vey impotant featue as summaized in Table. lthough the Tow Thomas cicuit has vey low passive sensitivities to all passive cicuit components, it suffes fom a athe dastic Qfacto enhancement effect due to the op amp finite gain bandwidth. Patialphase compensation which educes the phase lag aound the loop to half its value by just changing one connection has been discussed with simulation esults. Passive and active compensation methods to impove the cicuit pefomance fo highq designs ae summaized. Spice simulation esults ae included. The passive and active compensation simulation esults agee well with the ideal esponses as seen fom the simulation esults. cknowledgment The autho thanks the eviewes fo thei useful comments. efeences. J. Tow, step by step active filte design, IEEE Spectum 6 (969) 64 68. 2. L. Thomas, The biquad: Pat I Some pactical design consideations, IEEE Tans. icuit Theo. T8 (97) 350 357.
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