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J. of Active and Passive Electronic Devices, Vol. 5, pp. 321–326 Reprints available directly from the publisher Photocopying permitted by license only ©2010 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group Low Sensitivity Third Order Lowpass Butterworth Filter Using CFA R. Nandi1,*, Mousiki Kar1 and Soumik Das2 Department of Electronics & Telecommunication EngineeringJadavpur University, Kolkata-700032, India 2 Heritage Insitute of Technology, Kolkata-700107, India 1 A third−order lowpass (LP) Butterworth filter design scheme using two CFA devices is presented. The analysis is carried out utilising the parasitic and transadmittance poles of the AD−844 type CFA elements. The circuit is practically active−insensitive with respect to the port mismatch errors (∈) of the devices. The filter characteristic had been satisfactorily verified with both PSPICE macromodel simulation and hardware circuit implementation for a frequency range 6 MHz < fo < 30 MHz. Keywords: Butterworth filter, lowpass filter, third-order filter, CFA 1 INTRODUCTION The current feedback amplifier (CFA) device is now well−introduced in the literature as an active building block [1,2] that has some advantageous features over the conventional voltage operational amplifier (VOA) [3]. The significant feature of the CFA is its extended bandwidth (BW) capability that is practically independent of the closed loop gain [4]. This makes the CFA a superior choice over the VOA for the analog signal processing and filter design at relatively high frequencies. In the recent past, various CFA-based designs for first order and second order selective circuits were proposed [5−11]; some of these schemes assumed the parasitic/transadmittance capacitors to be very low and hence were neglected leading to relatively low frequency designs covering upto a few hundred of KHz. However a third−order Butterworth filter design using the CFA had not yet been reported, albeit other designs using VOA [12] and CC II are available [13]. *Corresponding author: [email protected], [email protected] 321 321-326 pp RC08-17 (Nandi).indd 321 6/11/2010 1:46:31 PM 322 R. Nandi et al. We consider here a practical CFA model [4,7,11] having parasitic elements at the non-inverting input (y) node and transadmittance components at the current source compensation (z) node; both of these appear as grounded shunt−RC arms. Our design approach here is to realise a third order maximally flat response LP−Butterworth filter with all the time constants normalised to equal value, while taking into account the device internal admittance elements. In order to equalise these internal capacitors (Cy,z), we added shunt external capacitors of appropriate value. The active sensitivity of the filter parameters (ωo, Q) with respect to device port−tracking errors (∈) had been shown to be extremely low. The analysis had been supported by some experimental results; the maximally flat third order LP response was verified for the range 6 MHz < f0 < 30 MHz. with both PSPICE macromodel simulation and hardware breadboard circuit implementation employing commercially available CFA devices (AD-844). 2 Analysis The proposed CFA−based circuit is shown in Fig. 1 where the device port relations are iz = ± (1 − ∈i)ix, vx = (1 − ∈v)vy and vo = (1 − ∈o)vz. The port mismatch errors (∈) are [10,11,14] usually extremely low |∈| < 0.03, and they vanish for an ideal device; initially we assume ∈ = 0. For the analysis we denote the device internal conductances by ‘g’ and external passive conductances by ‘G’. Writing iz = ± ix for a positive/negative polarity CFA device, we get the transfer F(s) = Eo/Ei in the form F(s) = Ei K d 3s + d 2 s2 + d1s + 1 (1) 3 x R1 vo −CFA y R2 y z Eo +CFA x z C1 R0 C3 R3 C2 Figure 1 CFA-based third-order filter. 321-326 pp RC08-17 (Nandi).indd 322 6/11/2010 1:46:31 PM Low Sensitivity Third Order Lowpass Butterworth Filter Using CFA 323 where K = Go/Gz2t d1 = [{b Cyt + (1 + n) Cz1t }/G1 }] + (Cz2t/Gz2t) (2) d2 = [Cz1t Cyt + p Cz2t { b Cyt + (1 + n) Cz1t}]/G1 G2 (3) d3 = Cz1t Cz2t Cyt/G1 G2 Gz2t (4) b = gz1/G2; n = gt/G1; p = G2/Gz2t (5) Cz1t = C1 + Cz1; Cz2t = Cz2 + C3; Cyt = Cy1 + Cy2 + C2 (6) and Gz2t = gz2 + G3; gt = gy1 + gy2 (7) Here ry,z (=1/gy,z) and Cy,z denote the device internal elements; as per data sheet [16] these values are Cy ≈ 3.5 pF, ry ≈ 3 M.ohm and Cz ≈ 6 pF, rz ≈ 5 M.ohm. Since we intended to design the filter at MHz ranges, R1,2,3 (=1/ G1,2,3) were selected in K.ohm ranges, which implies b, n << 1 and, p ≈ G2/ G3. Also we equalised Cz1t = Cz2t = Cyt = C by taking C1 = C - Cz1 , C2 = C − (Cy1 + Cy2) and C3 = C − Cz2. For a typical value of C = 8 pF (say), we set C1,3 = 2 pF and C2 = 1 pF. Further one may consider a simplified design with R1,2,3 = R. With these design approach, the denominator coefficients reduce to d1 = d2 = 2, and assuming the time constant is normalised to unity [12], we get F(s) = 1/{s3 + 2s2 + 2s + 1} (8) which is a third order LP Butterworth filter function. 3 Sensitivity Re−analysis of Fig. 1 with finite tracking errors (∈ ≠ 0) yields the modified coefficients as đ3 = (1 + ∈t1); đ2 = 2 (1 + ∈t1); đ1 = (2 + ∈t1) (9) Ќ = (1 + ∈t1 - ∈t2) (10) where ∈t1,2 = ∈i1,2 + ∈v1,2 + ∈o1,2; product of errors are neglected since |∈| << 1. 321-326 pp RC08-17 (Nandi).indd 323 6/11/2010 1:46:32 PM 324 R. Nandi et al. The sensitivities of the filter parameter are derived using Smu = (∂m/m)/ (∂u/u), and considering [17], we may write Sωu o = Sdu1 − Sdu2 and SQu = Sdu3 − Sdu2 − Sdu1 The sensitivity figures are listed below in Table-1 where µ = (1 + ∈t1). It may be seen that the proposed circuit is practically active-insensitive wherein the sensitivities,in general, may be summarised as − (S∈i,v.o 1)Q ≈ ∈i,v.o 1 << 1 (S∈i,v.o 2 )Q = 0 = (S∈i,v.o 1,2)ωo 4 Experimental Result The proposed circuit had been simulated with PSPICE macromodel data of the AD−844 device. The negative polarity device had been implemented by cascading a high output impedance current mirror [14] with the CFA. The components selected were R = 800 Ω and C = 8 pF ; the CFAs were supplied with 0 + 9 V regulated d.c. bias. A satisfactory maximally−flat LP Butterworth filter response at fo = 25.3 MHz as shown in Fig. 2 had been obtained. 5 Conclusion A new CFA−based circuit for realising third−order lowpass Butterworth filter is presented. The analysis is carried out utilising the parasitic capacitors of Table 1 Sensitivity characteristics of the Butterworth filter. m d1 d2 d3 ωo Q ∈i1 ∈i1/µ ∈i1/µ ∈i1/µ 0 − ∈i1/µ ∈v1 ∈v1/µ ∈v1/µ ∈v1/µ 0 − ∈v1/µ ∈o1 ∈o1/µ ∈o1/µ ∈o1/µ 0 − ∈o1/µ u ∈i2 0 0 0 0 0 ∈v2 0 0 0 0 0 ∈ o2 0 0 0 0 0 321-326 pp RC08-17 (Nandi).indd 324 6/11/2010 1:46:32 PM Low Sensitivity Third Order Lowpass Butterworth Filter Using CFA 325 0 -5 -10 dB -15 -20 -25 -30 1 10 100 MHz Figure 2 Experimental results on maximally flat third-order filter response at fo = 25.3 MHz using C (equalized value) = 8 pF; C1, 3 = 2 pF; C2 = 1 pF; R1,2,3 = 800Ω. by simulation : ______________ by hardware implementation : ______________ the high−frequency model of the AD−844 CFA device; the passive−RC components of the circuit had then been approximately chosen to obtain a normalised filter function [12]. The effect of rx (≈ 40Ω) at the x−node of the device had been neglected since rx can be virtually eleminated in some improved [4] CFA−models. The proposed topology would be suitable for IC adaptation since all capacitors are grounded and the circuit is practically active−insensitive. PSPICE macromodel simulation as well as hardware circuit tests verified satisfactorily the maximally−flat third-order lowpass Butterworth response for a range of select frequencies 6 MHz < fo < 30 MHz. It may be mentioned in conclusion that improved versions of the CFA are now being reported in the literature [18,19]; some of these devices, viz, 0PA-695 IC, provide wide bandwidth (∼ 450 MHz ) at very high slewrate (∼ 4.3 KV/µ.sec). Use of such a device would enhance the operating range of the proposed Butterworth filter with a practically active-insensitive feature. References [1] S. Franco. (1989). “Current feedback amplifiers”, Comlinear Corpn., Application Notes. 3–8. [2] C. Tomazou and J. Lidgey. 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[15] National Semiconductor Databook: Linear Applications.: A fast integrated voltage follower with low input current, p. AN5–1, 1977. [16] Analog Devices: Linear Products Databook., Norwood, MA., USA, 1990. [17] M. A. Soderstrand and S. K. Mitra. (1972). “Sensitivity analysis of third filters”, Int. Jl. Electronics, 30, 265–272. [18] B. Maundy, A. R. Sarkar and S. J. G Gift. (2006). “A new design topology for low voltage CMOS current feedback amplifiers”, IEEE Trans. Cct. Syst. CAS(II), 53, 34–38. [19] News Updates: Global Signal Processing Times, Internet version, Jan. 2004. 321-326 pp RC08-17 (Nandi).indd 326 6/11/2010 1:46:33 PM